Properties

Label 14-825e7-1.1-c5e7-0-2
Degree $14$
Conductor $2.601\times 10^{20}$
Sign $-1$
Analytic cond. $7.10070\times 10^{14}$
Root an. cond. $11.5028$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $7$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s + 63·3-s − 24·4-s − 567·6-s − 65·7-s + 486·8-s + 2.26e3·9-s + 847·11-s − 1.51e3·12-s + 113·13-s + 585·14-s − 1.13e3·16-s − 1.03e3·17-s − 2.04e4·18-s − 3.80e3·19-s − 4.09e3·21-s − 7.62e3·22-s − 514·23-s + 3.06e4·24-s − 1.01e3·26-s + 6.12e4·27-s + 1.56e3·28-s − 2.69e3·29-s − 1.72e4·31-s − 6.97e3·32-s + 5.33e4·33-s + 9.27e3·34-s + ⋯
L(s)  = 1  − 1.59·2-s + 4.04·3-s − 3/4·4-s − 6.42·6-s − 0.501·7-s + 2.68·8-s + 28/3·9-s + 2.11·11-s − 3.03·12-s + 0.185·13-s + 0.797·14-s − 1.11·16-s − 0.864·17-s − 14.8·18-s − 2.41·19-s − 2.02·21-s − 3.35·22-s − 0.202·23-s + 10.8·24-s − 0.295·26-s + 16.1·27-s + 0.376·28-s − 0.595·29-s − 3.22·31-s − 1.20·32-s + 8.52·33-s + 1.37·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(3^{7} \cdot 5^{14} \cdot 11^{7}\)
Sign: $-1$
Analytic conductor: \(7.10070\times 10^{14}\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 3^{7} \cdot 5^{14} \cdot 11^{7} ,\ ( \ : [5/2]^{7} ),\ -1 )\)

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 \( ( 1 - p^{2} T )^{7} \)
5 \( 1 \)
11 \( ( 1 - p^{2} T )^{7} \)
good2 \( 1 + 9 T + 105 T^{2} + 675 T^{3} + 2679 p T^{4} + 7649 p^{2} T^{5} + 24607 p^{3} T^{6} + 127 p^{13} T^{7} + 24607 p^{8} T^{8} + 7649 p^{12} T^{9} + 2679 p^{16} T^{10} + 675 p^{20} T^{11} + 105 p^{25} T^{12} + 9 p^{30} T^{13} + p^{35} T^{14} \)
7 \( 1 + 65 T + 42367 T^{2} + 5136020 T^{3} + 1145529576 T^{4} + 172016350840 T^{5} + 23032157085430 T^{6} + 3705419186848874 T^{7} + 23032157085430 p^{5} T^{8} + 172016350840 p^{10} T^{9} + 1145529576 p^{15} T^{10} + 5136020 p^{20} T^{11} + 42367 p^{25} T^{12} + 65 p^{30} T^{13} + p^{35} T^{14} \)
13 \( 1 - 113 T + 1809586 T^{2} - 61033581 T^{3} + 1554162355852 T^{4} + 8087016402861 T^{5} + 843388395257205997 T^{6} + 11566112020295245938 T^{7} + 843388395257205997 p^{5} T^{8} + 8087016402861 p^{10} T^{9} + 1554162355852 p^{15} T^{10} - 61033581 p^{20} T^{11} + 1809586 p^{25} T^{12} - 113 p^{30} T^{13} + p^{35} T^{14} \)
17 \( 1 + 1030 T + 7281445 T^{2} + 3868583408 T^{3} + 21243312682500 T^{4} + 3741306917492056 T^{5} + 37717035366340875762 T^{6} + \)\(10\!\cdots\!84\)\( T^{7} + 37717035366340875762 p^{5} T^{8} + 3741306917492056 p^{10} T^{9} + 21243312682500 p^{15} T^{10} + 3868583408 p^{20} T^{11} + 7281445 p^{25} T^{12} + 1030 p^{30} T^{13} + p^{35} T^{14} \)
19 \( 1 + 3803 T + 19562070 T^{2} + 50319854713 T^{3} + 148200073355655 T^{4} + 285939315848588586 T^{5} + \)\(60\!\cdots\!53\)\( T^{6} + \)\(91\!\cdots\!21\)\( T^{7} + \)\(60\!\cdots\!53\)\( p^{5} T^{8} + 285939315848588586 p^{10} T^{9} + 148200073355655 p^{15} T^{10} + 50319854713 p^{20} T^{11} + 19562070 p^{25} T^{12} + 3803 p^{30} T^{13} + p^{35} T^{14} \)
23 \( 1 + 514 T + 15007034 T^{2} + 1163046738 p T^{3} + 156540471582131 T^{4} + 338986231089844100 T^{5} + \)\(11\!\cdots\!01\)\( T^{6} + \)\(29\!\cdots\!98\)\( T^{7} + \)\(11\!\cdots\!01\)\( p^{5} T^{8} + 338986231089844100 p^{10} T^{9} + 156540471582131 p^{15} T^{10} + 1163046738 p^{21} T^{11} + 15007034 p^{25} T^{12} + 514 p^{30} T^{13} + p^{35} T^{14} \)
29 \( 1 + 2698 T + 102048646 T^{2} + 182525647862 T^{3} + 4648818068180928 T^{4} + 5142084748253253598 T^{5} + \)\(13\!\cdots\!29\)\( T^{6} + \)\(10\!\cdots\!84\)\( T^{7} + \)\(13\!\cdots\!29\)\( p^{5} T^{8} + 5142084748253253598 p^{10} T^{9} + 4648818068180928 p^{15} T^{10} + 182525647862 p^{20} T^{11} + 102048646 p^{25} T^{12} + 2698 p^{30} T^{13} + p^{35} T^{14} \)
31 \( 1 + 17233 T + 294577584 T^{2} + 3062891912441 T^{3} + 30055339945951858 T^{4} + \)\(22\!\cdots\!91\)\( T^{5} + \)\(15\!\cdots\!71\)\( T^{6} + \)\(84\!\cdots\!42\)\( T^{7} + \)\(15\!\cdots\!71\)\( p^{5} T^{8} + \)\(22\!\cdots\!91\)\( p^{10} T^{9} + 30055339945951858 p^{15} T^{10} + 3062891912441 p^{20} T^{11} + 294577584 p^{25} T^{12} + 17233 p^{30} T^{13} + p^{35} T^{14} \)
37 \( 1 + 23182 T + 593794213 T^{2} + 8502183622876 T^{3} + 127621887888662648 T^{4} + 36523767606417625596 p T^{5} + \)\(14\!\cdots\!46\)\( T^{6} + \)\(12\!\cdots\!76\)\( T^{7} + \)\(14\!\cdots\!46\)\( p^{5} T^{8} + 36523767606417625596 p^{11} T^{9} + 127621887888662648 p^{15} T^{10} + 8502183622876 p^{20} T^{11} + 593794213 p^{25} T^{12} + 23182 p^{30} T^{13} + p^{35} T^{14} \)
41 \( 1 + 16158 T + 453139329 T^{2} + 4993054022584 T^{3} + 69932687872165888 T^{4} + \)\(41\!\cdots\!20\)\( T^{5} + \)\(48\!\cdots\!02\)\( T^{6} + \)\(12\!\cdots\!36\)\( T^{7} + \)\(48\!\cdots\!02\)\( p^{5} T^{8} + \)\(41\!\cdots\!20\)\( p^{10} T^{9} + 69932687872165888 p^{15} T^{10} + 4993054022584 p^{20} T^{11} + 453139329 p^{25} T^{12} + 16158 p^{30} T^{13} + p^{35} T^{14} \)
43 \( 1 - 4249 T + 448845724 T^{2} - 1166793366105 T^{3} + 98506472123298662 T^{4} - \)\(16\!\cdots\!51\)\( T^{5} + \)\(17\!\cdots\!67\)\( T^{6} - \)\(24\!\cdots\!82\)\( T^{7} + \)\(17\!\cdots\!67\)\( p^{5} T^{8} - \)\(16\!\cdots\!51\)\( p^{10} T^{9} + 98506472123298662 p^{15} T^{10} - 1166793366105 p^{20} T^{11} + 448845724 p^{25} T^{12} - 4249 p^{30} T^{13} + p^{35} T^{14} \)
47 \( 1 + 7580 T + 1261398823 T^{2} + 7723296971660 T^{3} + 734956142025005208 T^{4} + \)\(37\!\cdots\!32\)\( T^{5} + \)\(25\!\cdots\!70\)\( T^{6} + \)\(10\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!70\)\( p^{5} T^{8} + \)\(37\!\cdots\!32\)\( p^{10} T^{9} + 734956142025005208 p^{15} T^{10} + 7723296971660 p^{20} T^{11} + 1261398823 p^{25} T^{12} + 7580 p^{30} T^{13} + p^{35} T^{14} \)
53 \( 1 + 20574 T + 984729355 T^{2} + 11625250031332 T^{3} + 282867128854717077 T^{4} + \)\(28\!\cdots\!30\)\( T^{5} - \)\(11\!\cdots\!33\)\( T^{6} - \)\(10\!\cdots\!72\)\( T^{7} - \)\(11\!\cdots\!33\)\( p^{5} T^{8} + \)\(28\!\cdots\!30\)\( p^{10} T^{9} + 282867128854717077 p^{15} T^{10} + 11625250031332 p^{20} T^{11} + 984729355 p^{25} T^{12} + 20574 p^{30} T^{13} + p^{35} T^{14} \)
59 \( 1 + 364 T + 2100658159 T^{2} + 12755018178636 T^{3} + 1713573225129985364 T^{4} + \)\(37\!\cdots\!56\)\( T^{5} + \)\(82\!\cdots\!26\)\( T^{6} + \)\(41\!\cdots\!88\)\( T^{7} + \)\(82\!\cdots\!26\)\( p^{5} T^{8} + \)\(37\!\cdots\!56\)\( p^{10} T^{9} + 1713573225129985364 p^{15} T^{10} + 12755018178636 p^{20} T^{11} + 2100658159 p^{25} T^{12} + 364 p^{30} T^{13} + p^{35} T^{14} \)
61 \( 1 + 28127 T + 4404961567 T^{2} + 102805675407078 T^{3} + 9045963811240026433 T^{4} + \)\(17\!\cdots\!93\)\( T^{5} + \)\(18\!\cdots\!27\)\( p T^{6} + \)\(18\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!27\)\( p^{6} T^{8} + \)\(17\!\cdots\!93\)\( p^{10} T^{9} + 9045963811240026433 p^{15} T^{10} + 102805675407078 p^{20} T^{11} + 4404961567 p^{25} T^{12} + 28127 p^{30} T^{13} + p^{35} T^{14} \)
67 \( 1 - 21493 T + 4533132489 T^{2} - 121567998667286 T^{3} + 11965793067420840153 T^{4} - \)\(34\!\cdots\!59\)\( T^{5} + \)\(21\!\cdots\!45\)\( T^{6} - \)\(58\!\cdots\!80\)\( T^{7} + \)\(21\!\cdots\!45\)\( p^{5} T^{8} - \)\(34\!\cdots\!59\)\( p^{10} T^{9} + 11965793067420840153 p^{15} T^{10} - 121567998667286 p^{20} T^{11} + 4533132489 p^{25} T^{12} - 21493 p^{30} T^{13} + p^{35} T^{14} \)
71 \( 1 + 177084 T + 21861181258 T^{2} + 1974993411311332 T^{3} + \)\(14\!\cdots\!31\)\( T^{4} + \)\(90\!\cdots\!92\)\( T^{5} + \)\(48\!\cdots\!93\)\( T^{6} + \)\(21\!\cdots\!68\)\( T^{7} + \)\(48\!\cdots\!93\)\( p^{5} T^{8} + \)\(90\!\cdots\!92\)\( p^{10} T^{9} + \)\(14\!\cdots\!31\)\( p^{15} T^{10} + 1974993411311332 p^{20} T^{11} + 21861181258 p^{25} T^{12} + 177084 p^{30} T^{13} + p^{35} T^{14} \)
73 \( 1 + 78670 T + 9788109595 T^{2} + 531681875083548 T^{3} + 41718249805428206833 T^{4} + \)\(18\!\cdots\!90\)\( T^{5} + \)\(11\!\cdots\!67\)\( T^{6} + \)\(46\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!67\)\( p^{5} T^{8} + \)\(18\!\cdots\!90\)\( p^{10} T^{9} + 41718249805428206833 p^{15} T^{10} + 531681875083548 p^{20} T^{11} + 9788109595 p^{25} T^{12} + 78670 p^{30} T^{13} + p^{35} T^{14} \)
79 \( 1 + 187432 T + 31099138199 T^{2} + 3367441554043048 T^{3} + \)\(32\!\cdots\!00\)\( T^{4} + \)\(25\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!38\)\( T^{6} + \)\(10\!\cdots\!56\)\( T^{7} + \)\(17\!\cdots\!38\)\( p^{5} T^{8} + \)\(25\!\cdots\!92\)\( p^{10} T^{9} + \)\(32\!\cdots\!00\)\( p^{15} T^{10} + 3367441554043048 p^{20} T^{11} + 31099138199 p^{25} T^{12} + 187432 p^{30} T^{13} + p^{35} T^{14} \)
83 \( 1 - 44592 T - 2044676734 T^{2} - 529986856032698 T^{3} + 50669201900733312648 T^{4} + \)\(86\!\cdots\!76\)\( T^{5} + \)\(56\!\cdots\!07\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!07\)\( p^{5} T^{8} + \)\(86\!\cdots\!76\)\( p^{10} T^{9} + 50669201900733312648 p^{15} T^{10} - 529986856032698 p^{20} T^{11} - 2044676734 p^{25} T^{12} - 44592 p^{30} T^{13} + p^{35} T^{14} \)
89 \( 1 + 151168 T + 43136796716 T^{2} + 4612755134627874 T^{3} + \)\(73\!\cdots\!66\)\( T^{4} + \)\(59\!\cdots\!76\)\( T^{5} + \)\(67\!\cdots\!05\)\( T^{6} + \)\(43\!\cdots\!64\)\( T^{7} + \)\(67\!\cdots\!05\)\( p^{5} T^{8} + \)\(59\!\cdots\!76\)\( p^{10} T^{9} + \)\(73\!\cdots\!66\)\( p^{15} T^{10} + 4612755134627874 p^{20} T^{11} + 43136796716 p^{25} T^{12} + 151168 p^{30} T^{13} + p^{35} T^{14} \)
97 \( 1 - 55589 T + 27771990688 T^{2} - 2483644230855539 T^{3} + \)\(41\!\cdots\!05\)\( T^{4} - \)\(45\!\cdots\!82\)\( T^{5} + \)\(43\!\cdots\!97\)\( T^{6} - \)\(50\!\cdots\!03\)\( T^{7} + \)\(43\!\cdots\!97\)\( p^{5} T^{8} - \)\(45\!\cdots\!82\)\( p^{10} T^{9} + \)\(41\!\cdots\!05\)\( p^{15} T^{10} - 2483644230855539 p^{20} T^{11} + 27771990688 p^{25} T^{12} - 55589 p^{30} T^{13} + p^{35} T^{14} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.38674334546592457614527094584, −4.37015481105268111985262997330, −4.31484913506695765515812008384, −4.22580938517385918341588016687, −3.66627820621278471982509115686, −3.66333641223519573829296696010, −3.60905552699983414160064663086, −3.56601046268762620698431812613, −3.55307316552815280205947802213, −3.36685248971807676564235058852, −3.06080289600461421282673557320, −2.84914741973679579029776514844, −2.76579645175307241934709950351, −2.71737279796770158005465484361, −2.27456759373651103897620473352, −2.19342892276920978944989755982, −2.08936981584469315437627378977, −1.91225835782658919833327896581, −1.87827805639411217032559007564, −1.54735028653630171891955200887, −1.36082101776373264380946692242, −1.30801600907136062872419968018, −1.23480981603752261937894463893, −1.21691700057161989784813509351, −0.992829614442473411000210559796, 0, 0, 0, 0, 0, 0, 0, 0.992829614442473411000210559796, 1.21691700057161989784813509351, 1.23480981603752261937894463893, 1.30801600907136062872419968018, 1.36082101776373264380946692242, 1.54735028653630171891955200887, 1.87827805639411217032559007564, 1.91225835782658919833327896581, 2.08936981584469315437627378977, 2.19342892276920978944989755982, 2.27456759373651103897620473352, 2.71737279796770158005465484361, 2.76579645175307241934709950351, 2.84914741973679579029776514844, 3.06080289600461421282673557320, 3.36685248971807676564235058852, 3.55307316552815280205947802213, 3.56601046268762620698431812613, 3.60905552699983414160064663086, 3.66333641223519573829296696010, 3.66627820621278471982509115686, 4.22580938517385918341588016687, 4.31484913506695765515812008384, 4.37015481105268111985262997330, 4.38674334546592457614527094584

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

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