Properties

Label 22-177e11-1.1-c5e11-0-0
Degree $22$
Conductor $5.342\times 10^{24}$
Sign $-1$
Analytic cond. $9.64877\times 10^{15}$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $11$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 99·3-s − 83·4-s − 192·5-s − 594·6-s − 371·7-s + 427·8-s + 5.34e3·9-s + 1.15e3·10-s − 698·11-s − 8.21e3·12-s − 1.55e3·13-s + 2.22e3·14-s − 1.90e4·15-s + 2.32e3·16-s − 4.79e3·17-s − 3.20e4·18-s − 3.75e3·19-s + 1.59e4·20-s − 3.67e4·21-s + 4.18e3·22-s − 7.32e3·23-s + 4.22e4·24-s + 5.17e3·25-s + 9.33e3·26-s + 2.08e5·27-s + 3.07e4·28-s + ⋯
L(s)  = 1  − 1.06·2-s + 6.35·3-s − 2.59·4-s − 3.43·5-s − 6.73·6-s − 2.86·7-s + 2.35·8-s + 22·9-s + 3.64·10-s − 1.73·11-s − 16.4·12-s − 2.55·13-s + 3.03·14-s − 21.8·15-s + 2.27·16-s − 4.02·17-s − 23.3·18-s − 2.38·19-s + 8.90·20-s − 18.1·21-s + 1.84·22-s − 2.88·23-s + 14.9·24-s + 1.65·25-s + 2.70·26-s + 55.0·27-s + 7.42·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{11} \cdot 59^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{11} \cdot 59^{11}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(22\)
Conductor: \(3^{11} \cdot 59^{11}\)
Sign: $-1$
Analytic conductor: \(9.64877\times 10^{15}\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(11\)
Selberg data: \((22,\ 3^{11} \cdot 59^{11} ,\ ( \ : [5/2]^{11} ),\ -1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - p^{2} T )^{11} \)
59 \( ( 1 - p^{2} T )^{11} \)
good2 \( 1 + 3 p T + 119 T^{2} + 785 T^{3} + 303 p^{5} T^{4} + 61613 T^{5} + 145509 p^{2} T^{6} + 108259 p^{5} T^{7} + 1736865 p^{4} T^{8} + 4776973 p^{5} T^{9} + 1056929 p^{10} T^{10} + 84137505 p^{6} T^{11} + 1056929 p^{15} T^{12} + 4776973 p^{15} T^{13} + 1736865 p^{19} T^{14} + 108259 p^{25} T^{15} + 145509 p^{27} T^{16} + 61613 p^{30} T^{17} + 303 p^{40} T^{18} + 785 p^{40} T^{19} + 119 p^{45} T^{20} + 3 p^{51} T^{21} + p^{55} T^{22} \)
5 \( 1 + 192 T + 31686 T^{2} + 742888 p T^{3} + 360597611 T^{4} + 5825416596 p T^{5} + 1982523021341 T^{6} + 4531057687856 p^{2} T^{7} + 5423641433146986 T^{8} + 42377419464341208 p T^{9} + 57265622722252979 p^{3} T^{10} + \)\(29\!\cdots\!48\)\( T^{11} + 57265622722252979 p^{8} T^{12} + 42377419464341208 p^{11} T^{13} + 5423641433146986 p^{15} T^{14} + 4531057687856 p^{22} T^{15} + 1982523021341 p^{25} T^{16} + 5825416596 p^{31} T^{17} + 360597611 p^{35} T^{18} + 742888 p^{41} T^{19} + 31686 p^{45} T^{20} + 192 p^{50} T^{21} + p^{55} T^{22} \)
7 \( 1 + 53 p T + 166420 T^{2} + 6263746 p T^{3} + 12162403712 T^{4} + 2544492780163 T^{5} + 77218270385286 p T^{6} + 94502435023499829 T^{7} + 16573083801901627156 T^{8} + \)\(24\!\cdots\!70\)\( T^{9} + \)\(37\!\cdots\!07\)\( T^{10} + \)\(48\!\cdots\!46\)\( T^{11} + \)\(37\!\cdots\!07\)\( p^{5} T^{12} + \)\(24\!\cdots\!70\)\( p^{10} T^{13} + 16573083801901627156 p^{15} T^{14} + 94502435023499829 p^{20} T^{15} + 77218270385286 p^{26} T^{16} + 2544492780163 p^{30} T^{17} + 12162403712 p^{35} T^{18} + 6263746 p^{41} T^{19} + 166420 p^{45} T^{20} + 53 p^{51} T^{21} + p^{55} T^{22} \)
11 \( 1 + 698 T + 842159 T^{2} + 282798894 T^{3} + 221347866805 T^{4} + 2790143706552 p T^{5} + 43414241573905689 T^{6} + 4219816194278589834 T^{7} + \)\(85\!\cdots\!25\)\( p T^{8} + \)\(12\!\cdots\!46\)\( T^{9} + \)\(12\!\cdots\!77\)\( T^{10} - \)\(15\!\cdots\!16\)\( T^{11} + \)\(12\!\cdots\!77\)\( p^{5} T^{12} + \)\(12\!\cdots\!46\)\( p^{10} T^{13} + \)\(85\!\cdots\!25\)\( p^{16} T^{14} + 4219816194278589834 p^{20} T^{15} + 43414241573905689 p^{25} T^{16} + 2790143706552 p^{31} T^{17} + 221347866805 p^{35} T^{18} + 282798894 p^{40} T^{19} + 842159 p^{45} T^{20} + 698 p^{50} T^{21} + p^{55} T^{22} \)
13 \( 1 + 1556 T + 2739913 T^{2} + 2924253880 T^{3} + 3057285307903 T^{4} + 2456187620998976 T^{5} + 1912863944234469739 T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(77\!\cdots\!29\)\( T^{8} + \)\(41\!\cdots\!20\)\( T^{9} + \)\(24\!\cdots\!13\)\( T^{10} + \)\(13\!\cdots\!64\)\( T^{11} + \)\(24\!\cdots\!13\)\( p^{5} T^{12} + \)\(41\!\cdots\!20\)\( p^{10} T^{13} + \)\(77\!\cdots\!29\)\( p^{15} T^{14} + \)\(12\!\cdots\!40\)\( p^{20} T^{15} + 1912863944234469739 p^{25} T^{16} + 2456187620998976 p^{30} T^{17} + 3057285307903 p^{35} T^{18} + 2924253880 p^{40} T^{19} + 2739913 p^{45} T^{20} + 1556 p^{50} T^{21} + p^{55} T^{22} \)
17 \( 1 + 4793 T + 18843964 T^{2} + 53952134734 T^{3} + 134569129524518 T^{4} + 286928670328145251 T^{5} + \)\(55\!\cdots\!72\)\( T^{6} + \)\(95\!\cdots\!61\)\( T^{7} + \)\(15\!\cdots\!78\)\( T^{8} + \)\(21\!\cdots\!10\)\( T^{9} + \)\(29\!\cdots\!95\)\( T^{10} + \)\(36\!\cdots\!58\)\( T^{11} + \)\(29\!\cdots\!95\)\( p^{5} T^{12} + \)\(21\!\cdots\!10\)\( p^{10} T^{13} + \)\(15\!\cdots\!78\)\( p^{15} T^{14} + \)\(95\!\cdots\!61\)\( p^{20} T^{15} + \)\(55\!\cdots\!72\)\( p^{25} T^{16} + 286928670328145251 p^{30} T^{17} + 134569129524518 p^{35} T^{18} + 53952134734 p^{40} T^{19} + 18843964 p^{45} T^{20} + 4793 p^{50} T^{21} + p^{55} T^{22} \)
19 \( 1 + 3753 T + 18560597 T^{2} + 50922044923 T^{3} + 163720060253515 T^{4} + 371240138717778678 T^{5} + \)\(93\!\cdots\!02\)\( T^{6} + \)\(18\!\cdots\!58\)\( T^{7} + \)\(39\!\cdots\!55\)\( T^{8} + \)\(67\!\cdots\!13\)\( T^{9} + \)\(12\!\cdots\!56\)\( T^{10} + \)\(18\!\cdots\!22\)\( T^{11} + \)\(12\!\cdots\!56\)\( p^{5} T^{12} + \)\(67\!\cdots\!13\)\( p^{10} T^{13} + \)\(39\!\cdots\!55\)\( p^{15} T^{14} + \)\(18\!\cdots\!58\)\( p^{20} T^{15} + \)\(93\!\cdots\!02\)\( p^{25} T^{16} + 371240138717778678 p^{30} T^{17} + 163720060253515 p^{35} T^{18} + 50922044923 p^{40} T^{19} + 18560597 p^{45} T^{20} + 3753 p^{50} T^{21} + p^{55} T^{22} \)
23 \( 1 + 7323 T + 49978469 T^{2} + 230971529573 T^{3} + 1046169483792175 T^{4} + 3949682037296415038 T^{5} + \)\(14\!\cdots\!66\)\( T^{6} + \)\(47\!\cdots\!86\)\( T^{7} + \)\(15\!\cdots\!59\)\( T^{8} + \)\(19\!\cdots\!13\)\( p T^{9} + \)\(12\!\cdots\!88\)\( T^{10} + \)\(31\!\cdots\!34\)\( T^{11} + \)\(12\!\cdots\!88\)\( p^{5} T^{12} + \)\(19\!\cdots\!13\)\( p^{11} T^{13} + \)\(15\!\cdots\!59\)\( p^{15} T^{14} + \)\(47\!\cdots\!86\)\( p^{20} T^{15} + \)\(14\!\cdots\!66\)\( p^{25} T^{16} + 3949682037296415038 p^{30} T^{17} + 1046169483792175 p^{35} T^{18} + 230971529573 p^{40} T^{19} + 49978469 p^{45} T^{20} + 7323 p^{50} T^{21} + p^{55} T^{22} \)
29 \( 1 + 15467 T + 208606977 T^{2} + 2078363510863 T^{3} + 624140881886717 p T^{4} + \)\(13\!\cdots\!48\)\( T^{5} + \)\(94\!\cdots\!48\)\( T^{6} + \)\(58\!\cdots\!36\)\( T^{7} + \)\(34\!\cdots\!31\)\( T^{8} + \)\(18\!\cdots\!29\)\( T^{9} + \)\(91\!\cdots\!14\)\( T^{10} + \)\(42\!\cdots\!06\)\( T^{11} + \)\(91\!\cdots\!14\)\( p^{5} T^{12} + \)\(18\!\cdots\!29\)\( p^{10} T^{13} + \)\(34\!\cdots\!31\)\( p^{15} T^{14} + \)\(58\!\cdots\!36\)\( p^{20} T^{15} + \)\(94\!\cdots\!48\)\( p^{25} T^{16} + \)\(13\!\cdots\!48\)\( p^{30} T^{17} + 624140881886717 p^{36} T^{18} + 2078363510863 p^{40} T^{19} + 208606977 p^{45} T^{20} + 15467 p^{50} T^{21} + p^{55} T^{22} \)
31 \( 1 + 5151 T + 143752753 T^{2} + 683499099413 T^{3} + 10354533648906829 T^{4} + 47221511563610523390 T^{5} + \)\(53\!\cdots\!90\)\( T^{6} + \)\(23\!\cdots\!22\)\( T^{7} + \)\(22\!\cdots\!93\)\( T^{8} + \)\(88\!\cdots\!43\)\( T^{9} + \)\(74\!\cdots\!96\)\( T^{10} + \)\(27\!\cdots\!50\)\( T^{11} + \)\(74\!\cdots\!96\)\( p^{5} T^{12} + \)\(88\!\cdots\!43\)\( p^{10} T^{13} + \)\(22\!\cdots\!93\)\( p^{15} T^{14} + \)\(23\!\cdots\!22\)\( p^{20} T^{15} + \)\(53\!\cdots\!90\)\( p^{25} T^{16} + 47221511563610523390 p^{30} T^{17} + 10354533648906829 p^{35} T^{18} + 683499099413 p^{40} T^{19} + 143752753 p^{45} T^{20} + 5151 p^{50} T^{21} + p^{55} T^{22} \)
37 \( 1 - 8623 T + 334046334 T^{2} - 2570567373072 T^{3} + 57022525589697024 T^{4} - \)\(42\!\cdots\!47\)\( T^{5} + \)\(70\!\cdots\!38\)\( T^{6} - \)\(51\!\cdots\!85\)\( T^{7} + \)\(69\!\cdots\!28\)\( T^{8} - \)\(49\!\cdots\!16\)\( T^{9} + \)\(57\!\cdots\!49\)\( T^{10} - \)\(38\!\cdots\!54\)\( T^{11} + \)\(57\!\cdots\!49\)\( p^{5} T^{12} - \)\(49\!\cdots\!16\)\( p^{10} T^{13} + \)\(69\!\cdots\!28\)\( p^{15} T^{14} - \)\(51\!\cdots\!85\)\( p^{20} T^{15} + \)\(70\!\cdots\!38\)\( p^{25} T^{16} - \)\(42\!\cdots\!47\)\( p^{30} T^{17} + 57022525589697024 p^{35} T^{18} - 2570567373072 p^{40} T^{19} + 334046334 p^{45} T^{20} - 8623 p^{50} T^{21} + p^{55} T^{22} \)
41 \( 1 + 6369 T + 790696708 T^{2} + 4374607593618 T^{3} + 307641457844497002 T^{4} + \)\(15\!\cdots\!15\)\( T^{5} + \)\(79\!\cdots\!32\)\( T^{6} + \)\(38\!\cdots\!45\)\( T^{7} + \)\(14\!\cdots\!62\)\( T^{8} + \)\(68\!\cdots\!70\)\( T^{9} + \)\(21\!\cdots\!71\)\( T^{10} + \)\(91\!\cdots\!38\)\( T^{11} + \)\(21\!\cdots\!71\)\( p^{5} T^{12} + \)\(68\!\cdots\!70\)\( p^{10} T^{13} + \)\(14\!\cdots\!62\)\( p^{15} T^{14} + \)\(38\!\cdots\!45\)\( p^{20} T^{15} + \)\(79\!\cdots\!32\)\( p^{25} T^{16} + \)\(15\!\cdots\!15\)\( p^{30} T^{17} + 307641457844497002 p^{35} T^{18} + 4374607593618 p^{40} T^{19} + 790696708 p^{45} T^{20} + 6369 p^{50} T^{21} + p^{55} T^{22} \)
43 \( 1 + 20506 T + 554571291 T^{2} + 4315090007402 T^{3} + 71076650924836065 T^{4} + \)\(11\!\cdots\!04\)\( T^{5} + \)\(85\!\cdots\!57\)\( T^{6} + \)\(33\!\cdots\!46\)\( T^{7} + \)\(14\!\cdots\!95\)\( T^{8} + \)\(72\!\cdots\!90\)\( T^{9} + \)\(20\!\cdots\!13\)\( T^{10} + \)\(14\!\cdots\!40\)\( T^{11} + \)\(20\!\cdots\!13\)\( p^{5} T^{12} + \)\(72\!\cdots\!90\)\( p^{10} T^{13} + \)\(14\!\cdots\!95\)\( p^{15} T^{14} + \)\(33\!\cdots\!46\)\( p^{20} T^{15} + \)\(85\!\cdots\!57\)\( p^{25} T^{16} + \)\(11\!\cdots\!04\)\( p^{30} T^{17} + 71076650924836065 p^{35} T^{18} + 4315090007402 p^{40} T^{19} + 554571291 p^{45} T^{20} + 20506 p^{50} T^{21} + p^{55} T^{22} \)
47 \( 1 + 47899 T + 2034808463 T^{2} + 53536227197069 T^{3} + 1335946687754377315 T^{4} + \)\(25\!\cdots\!98\)\( T^{5} + \)\(50\!\cdots\!78\)\( T^{6} + \)\(85\!\cdots\!26\)\( T^{7} + \)\(16\!\cdots\!07\)\( T^{8} + \)\(26\!\cdots\!07\)\( T^{9} + \)\(46\!\cdots\!62\)\( T^{10} + \)\(68\!\cdots\!26\)\( T^{11} + \)\(46\!\cdots\!62\)\( p^{5} T^{12} + \)\(26\!\cdots\!07\)\( p^{10} T^{13} + \)\(16\!\cdots\!07\)\( p^{15} T^{14} + \)\(85\!\cdots\!26\)\( p^{20} T^{15} + \)\(50\!\cdots\!78\)\( p^{25} T^{16} + \)\(25\!\cdots\!98\)\( p^{30} T^{17} + 1335946687754377315 p^{35} T^{18} + 53536227197069 p^{40} T^{19} + 2034808463 p^{45} T^{20} + 47899 p^{50} T^{21} + p^{55} T^{22} \)
53 \( 1 + 80048 T + 4009453302 T^{2} + 169677233481960 T^{3} + 6362487625968844379 T^{4} + \)\(21\!\cdots\!04\)\( T^{5} + \)\(64\!\cdots\!13\)\( T^{6} + \)\(18\!\cdots\!76\)\( T^{7} + \)\(46\!\cdots\!46\)\( T^{8} + \)\(11\!\cdots\!56\)\( T^{9} + \)\(25\!\cdots\!95\)\( T^{10} + \)\(53\!\cdots\!76\)\( T^{11} + \)\(25\!\cdots\!95\)\( p^{5} T^{12} + \)\(11\!\cdots\!56\)\( p^{10} T^{13} + \)\(46\!\cdots\!46\)\( p^{15} T^{14} + \)\(18\!\cdots\!76\)\( p^{20} T^{15} + \)\(64\!\cdots\!13\)\( p^{25} T^{16} + \)\(21\!\cdots\!04\)\( p^{30} T^{17} + 6362487625968844379 p^{35} T^{18} + 169677233481960 p^{40} T^{19} + 4009453302 p^{45} T^{20} + 80048 p^{50} T^{21} + p^{55} T^{22} \)
61 \( 1 + 82527 T + 9624515589 T^{2} + 580000894125419 T^{3} + 39369343598255542125 T^{4} + \)\(18\!\cdots\!72\)\( T^{5} + \)\(94\!\cdots\!40\)\( T^{6} + \)\(37\!\cdots\!92\)\( T^{7} + \)\(15\!\cdots\!55\)\( T^{8} + \)\(50\!\cdots\!85\)\( T^{9} + \)\(17\!\cdots\!34\)\( T^{10} + \)\(50\!\cdots\!82\)\( T^{11} + \)\(17\!\cdots\!34\)\( p^{5} T^{12} + \)\(50\!\cdots\!85\)\( p^{10} T^{13} + \)\(15\!\cdots\!55\)\( p^{15} T^{14} + \)\(37\!\cdots\!92\)\( p^{20} T^{15} + \)\(94\!\cdots\!40\)\( p^{25} T^{16} + \)\(18\!\cdots\!72\)\( p^{30} T^{17} + 39369343598255542125 p^{35} T^{18} + 580000894125419 p^{40} T^{19} + 9624515589 p^{45} T^{20} + 82527 p^{50} T^{21} + p^{55} T^{22} \)
67 \( 1 + 166976 T + 21146754554 T^{2} + 1925619803352400 T^{3} + \)\(15\!\cdots\!93\)\( T^{4} + \)\(10\!\cdots\!04\)\( T^{5} + \)\(60\!\cdots\!61\)\( T^{6} + \)\(32\!\cdots\!08\)\( T^{7} + \)\(15\!\cdots\!10\)\( T^{8} + \)\(70\!\cdots\!64\)\( T^{9} + \)\(29\!\cdots\!17\)\( T^{10} + \)\(11\!\cdots\!80\)\( T^{11} + \)\(29\!\cdots\!17\)\( p^{5} T^{12} + \)\(70\!\cdots\!64\)\( p^{10} T^{13} + \)\(15\!\cdots\!10\)\( p^{15} T^{14} + \)\(32\!\cdots\!08\)\( p^{20} T^{15} + \)\(60\!\cdots\!61\)\( p^{25} T^{16} + \)\(10\!\cdots\!04\)\( p^{30} T^{17} + \)\(15\!\cdots\!93\)\( p^{35} T^{18} + 1925619803352400 p^{40} T^{19} + 21146754554 p^{45} T^{20} + 166976 p^{50} T^{21} + p^{55} T^{22} \)
71 \( 1 + 183560 T + 28586808837 T^{2} + 3132367322901374 T^{3} + \)\(29\!\cdots\!63\)\( T^{4} + \)\(24\!\cdots\!34\)\( T^{5} + \)\(17\!\cdots\!53\)\( T^{6} + \)\(11\!\cdots\!34\)\( T^{7} + \)\(64\!\cdots\!63\)\( T^{8} + \)\(34\!\cdots\!00\)\( T^{9} + \)\(16\!\cdots\!23\)\( T^{10} + \)\(73\!\cdots\!96\)\( T^{11} + \)\(16\!\cdots\!23\)\( p^{5} T^{12} + \)\(34\!\cdots\!00\)\( p^{10} T^{13} + \)\(64\!\cdots\!63\)\( p^{15} T^{14} + \)\(11\!\cdots\!34\)\( p^{20} T^{15} + \)\(17\!\cdots\!53\)\( p^{25} T^{16} + \)\(24\!\cdots\!34\)\( p^{30} T^{17} + \)\(29\!\cdots\!63\)\( p^{35} T^{18} + 3132367322901374 p^{40} T^{19} + 28586808837 p^{45} T^{20} + 183560 p^{50} T^{21} + p^{55} T^{22} \)
73 \( 1 + 36809 T + 10609456769 T^{2} + 370936496353069 T^{3} + 62429520121314917079 T^{4} + \)\(21\!\cdots\!64\)\( T^{5} + \)\(25\!\cdots\!48\)\( T^{6} + \)\(86\!\cdots\!12\)\( T^{7} + \)\(81\!\cdots\!81\)\( T^{8} + \)\(26\!\cdots\!79\)\( T^{9} + \)\(20\!\cdots\!90\)\( T^{10} + \)\(60\!\cdots\!50\)\( T^{11} + \)\(20\!\cdots\!90\)\( p^{5} T^{12} + \)\(26\!\cdots\!79\)\( p^{10} T^{13} + \)\(81\!\cdots\!81\)\( p^{15} T^{14} + \)\(86\!\cdots\!12\)\( p^{20} T^{15} + \)\(25\!\cdots\!48\)\( p^{25} T^{16} + \)\(21\!\cdots\!64\)\( p^{30} T^{17} + 62429520121314917079 p^{35} T^{18} + 370936496353069 p^{40} T^{19} + 10609456769 p^{45} T^{20} + 36809 p^{50} T^{21} + p^{55} T^{22} \)
79 \( 1 + 281518 T + 57390761315 T^{2} + 8068239705886394 T^{3} + \)\(94\!\cdots\!09\)\( T^{4} + \)\(89\!\cdots\!92\)\( T^{5} + \)\(74\!\cdots\!89\)\( T^{6} + \)\(53\!\cdots\!34\)\( T^{7} + \)\(34\!\cdots\!15\)\( T^{8} + \)\(20\!\cdots\!98\)\( T^{9} + \)\(11\!\cdots\!53\)\( T^{10} + \)\(64\!\cdots\!08\)\( T^{11} + \)\(11\!\cdots\!53\)\( p^{5} T^{12} + \)\(20\!\cdots\!98\)\( p^{10} T^{13} + \)\(34\!\cdots\!15\)\( p^{15} T^{14} + \)\(53\!\cdots\!34\)\( p^{20} T^{15} + \)\(74\!\cdots\!89\)\( p^{25} T^{16} + \)\(89\!\cdots\!92\)\( p^{30} T^{17} + \)\(94\!\cdots\!09\)\( p^{35} T^{18} + 8068239705886394 p^{40} T^{19} + 57390761315 p^{45} T^{20} + 281518 p^{50} T^{21} + p^{55} T^{22} \)
83 \( 1 + 254691 T + 58214165812 T^{2} + 9072153820849260 T^{3} + \)\(12\!\cdots\!02\)\( T^{4} + \)\(14\!\cdots\!33\)\( T^{5} + \)\(15\!\cdots\!86\)\( T^{6} + \)\(14\!\cdots\!35\)\( T^{7} + \)\(12\!\cdots\!94\)\( T^{8} + \)\(97\!\cdots\!68\)\( T^{9} + \)\(69\!\cdots\!75\)\( T^{10} + \)\(45\!\cdots\!70\)\( T^{11} + \)\(69\!\cdots\!75\)\( p^{5} T^{12} + \)\(97\!\cdots\!68\)\( p^{10} T^{13} + \)\(12\!\cdots\!94\)\( p^{15} T^{14} + \)\(14\!\cdots\!35\)\( p^{20} T^{15} + \)\(15\!\cdots\!86\)\( p^{25} T^{16} + \)\(14\!\cdots\!33\)\( p^{30} T^{17} + \)\(12\!\cdots\!02\)\( p^{35} T^{18} + 9072153820849260 p^{40} T^{19} + 58214165812 p^{45} T^{20} + 254691 p^{50} T^{21} + p^{55} T^{22} \)
89 \( 1 + 89687 T + 41476178769 T^{2} + 3458236998583095 T^{3} + \)\(82\!\cdots\!91\)\( T^{4} + \)\(64\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!04\)\( T^{6} + \)\(76\!\cdots\!56\)\( T^{7} + \)\(98\!\cdots\!69\)\( T^{8} + \)\(65\!\cdots\!37\)\( T^{9} + \)\(70\!\cdots\!58\)\( T^{10} + \)\(42\!\cdots\!98\)\( T^{11} + \)\(70\!\cdots\!58\)\( p^{5} T^{12} + \)\(65\!\cdots\!37\)\( p^{10} T^{13} + \)\(98\!\cdots\!69\)\( p^{15} T^{14} + \)\(76\!\cdots\!56\)\( p^{20} T^{15} + \)\(10\!\cdots\!04\)\( p^{25} T^{16} + \)\(64\!\cdots\!68\)\( p^{30} T^{17} + \)\(82\!\cdots\!91\)\( p^{35} T^{18} + 3458236998583095 p^{40} T^{19} + 41476178769 p^{45} T^{20} + 89687 p^{50} T^{21} + p^{55} T^{22} \)
97 \( 1 + 45828 T + 46903421186 T^{2} + 4684898508660382 T^{3} + \)\(11\!\cdots\!65\)\( T^{4} + \)\(15\!\cdots\!52\)\( T^{5} + \)\(20\!\cdots\!07\)\( T^{6} + \)\(28\!\cdots\!66\)\( T^{7} + \)\(31\!\cdots\!34\)\( T^{8} + \)\(34\!\cdots\!00\)\( T^{9} + \)\(36\!\cdots\!51\)\( T^{10} + \)\(33\!\cdots\!56\)\( T^{11} + \)\(36\!\cdots\!51\)\( p^{5} T^{12} + \)\(34\!\cdots\!00\)\( p^{10} T^{13} + \)\(31\!\cdots\!34\)\( p^{15} T^{14} + \)\(28\!\cdots\!66\)\( p^{20} T^{15} + \)\(20\!\cdots\!07\)\( p^{25} T^{16} + \)\(15\!\cdots\!52\)\( p^{30} T^{17} + \)\(11\!\cdots\!65\)\( p^{35} T^{18} + 4684898508660382 p^{40} T^{19} + 46903421186 p^{45} T^{20} + 45828 p^{50} T^{21} + p^{55} T^{22} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.36042903086877353026528197734, −4.29053251326394826218770699554, −4.20809623814440702891570597989, −3.81018393958993599091467508011, −3.65704564083840324805145906435, −3.62941653775026072006100413011, −3.49060940722434141366489623569, −3.45290446205693656438848701835, −3.41499865368432585025667328071, −3.39295175934334592423867425948, −3.17362033918638656828068589940, −2.93966799828149235409535210629, −2.82636828125798405848457810768, −2.68499442050990804705934193820, −2.64416288601302314133943915513, −2.43163010351496519199752879567, −2.40083319562865224653682549025, −2.17623096942402966892730741968, −2.11915960750502431681863590638, −1.79010961321683077222608869498, −1.72548427306702597091183689824, −1.61857771753790420162371538831, −1.59982124300523980286507738238, −1.32676163948764000618424987288, −1.31209003978485581879843004447, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.31209003978485581879843004447, 1.32676163948764000618424987288, 1.59982124300523980286507738238, 1.61857771753790420162371538831, 1.72548427306702597091183689824, 1.79010961321683077222608869498, 2.11915960750502431681863590638, 2.17623096942402966892730741968, 2.40083319562865224653682549025, 2.43163010351496519199752879567, 2.64416288601302314133943915513, 2.68499442050990804705934193820, 2.82636828125798405848457810768, 2.93966799828149235409535210629, 3.17362033918638656828068589940, 3.39295175934334592423867425948, 3.41499865368432585025667328071, 3.45290446205693656438848701835, 3.49060940722434141366489623569, 3.62941653775026072006100413011, 3.65704564083840324805145906435, 3.81018393958993599091467508011, 4.20809623814440702891570597989, 4.29053251326394826218770699554, 4.36042903086877353026528197734

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.