Properties

Label 22-8030e11-1.1-c1e11-0-0
Degree $22$
Conductor $8.951\times 10^{42}$
Sign $-1$
Analytic cond. $7.53215\times 10^{19}$
Root an. cond. $8.00748$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $11$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 11·2-s − 5·3-s + 66·4-s + 11·5-s + 55·6-s − 7-s − 286·8-s + 2·9-s − 121·10-s + 11·11-s − 330·12-s − 8·13-s + 11·14-s − 55·15-s + 1.00e3·16-s − 15·17-s − 22·18-s + 5·19-s + 726·20-s + 5·21-s − 121·22-s − 24·23-s + 1.43e3·24-s + 66·25-s + 88·26-s + 26·27-s − 66·28-s + ⋯
L(s)  = 1  − 7.77·2-s − 2.88·3-s + 33·4-s + 4.91·5-s + 22.4·6-s − 0.377·7-s − 101.·8-s + 2/3·9-s − 38.2·10-s + 3.31·11-s − 95.2·12-s − 2.21·13-s + 2.93·14-s − 14.2·15-s + 250.·16-s − 3.63·17-s − 5.18·18-s + 1.14·19-s + 162.·20-s + 1.09·21-s − 25.7·22-s − 5.00·23-s + 291.·24-s + 66/5·25-s + 17.2·26-s + 5.00·27-s − 12.4·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 73^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 73^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(22\)
Conductor: \(2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 73^{11}\)
Sign: $-1$
Analytic conductor: \(7.53215\times 10^{19}\)
Root analytic conductor: \(8.00748\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(11\)
Selberg data: \((22,\ 2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 73^{11} ,\ ( \ : [1/2]^{11} ),\ -1 )\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{11} \)
5 \( ( 1 - T )^{11} \)
11 \( ( 1 - T )^{11} \)
73 \( ( 1 + T )^{11} \)
good3 \( 1 + 5 T + 23 T^{2} + 79 T^{3} + 253 T^{4} + 227 p T^{5} + 581 p T^{6} + 4000 T^{7} + 2917 p T^{8} + 17474 T^{9} + 33544 T^{10} + 59345 T^{11} + 33544 p T^{12} + 17474 p^{2} T^{13} + 2917 p^{4} T^{14} + 4000 p^{4} T^{15} + 581 p^{6} T^{16} + 227 p^{7} T^{17} + 253 p^{7} T^{18} + 79 p^{8} T^{19} + 23 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \)
7 \( 1 + T + 48 T^{2} + 19 T^{3} + 1118 T^{4} - 25 T^{5} + 17256 T^{6} - 4503 T^{7} + 198320 T^{8} - 69213 T^{9} + 1769843 T^{10} - 602510 T^{11} + 1769843 p T^{12} - 69213 p^{2} T^{13} + 198320 p^{3} T^{14} - 4503 p^{4} T^{15} + 17256 p^{5} T^{16} - 25 p^{6} T^{17} + 1118 p^{7} T^{18} + 19 p^{8} T^{19} + 48 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \)
13 \( 1 + 8 T + 102 T^{2} + 555 T^{3} + 4156 T^{4} + 17500 T^{5} + 103075 T^{6} + 376690 T^{7} + 1987822 T^{8} + 6743649 T^{9} + 32218269 T^{10} + 99151816 T^{11} + 32218269 p T^{12} + 6743649 p^{2} T^{13} + 1987822 p^{3} T^{14} + 376690 p^{4} T^{15} + 103075 p^{5} T^{16} + 17500 p^{6} T^{17} + 4156 p^{7} T^{18} + 555 p^{8} T^{19} + 102 p^{9} T^{20} + 8 p^{10} T^{21} + p^{11} T^{22} \)
17 \( 1 + 15 T + 206 T^{2} + 1940 T^{3} + 16527 T^{4} + 117992 T^{5} + 773078 T^{6} + 4512366 T^{7} + 24467025 T^{8} + 120995894 T^{9} + 559936829 T^{10} + 2383634442 T^{11} + 559936829 p T^{12} + 120995894 p^{2} T^{13} + 24467025 p^{3} T^{14} + 4512366 p^{4} T^{15} + 773078 p^{5} T^{16} + 117992 p^{6} T^{17} + 16527 p^{7} T^{18} + 1940 p^{8} T^{19} + 206 p^{9} T^{20} + 15 p^{10} T^{21} + p^{11} T^{22} \)
19 \( 1 - 5 T + 108 T^{2} - 612 T^{3} + 6254 T^{4} - 34244 T^{5} + 251486 T^{6} - 1236889 T^{7} + 7525775 T^{8} - 33417015 T^{9} + 175772376 T^{10} - 37470478 p T^{11} + 175772376 p T^{12} - 33417015 p^{2} T^{13} + 7525775 p^{3} T^{14} - 1236889 p^{4} T^{15} + 251486 p^{5} T^{16} - 34244 p^{6} T^{17} + 6254 p^{7} T^{18} - 612 p^{8} T^{19} + 108 p^{9} T^{20} - 5 p^{10} T^{21} + p^{11} T^{22} \)
23 \( 1 + 24 T + 442 T^{2} + 5828 T^{3} + 65310 T^{4} + 616202 T^{5} + 5147620 T^{6} + 38068536 T^{7} + 254260798 T^{8} + 1534931140 T^{9} + 8448417831 T^{10} + 42372421740 T^{11} + 8448417831 p T^{12} + 1534931140 p^{2} T^{13} + 254260798 p^{3} T^{14} + 38068536 p^{4} T^{15} + 5147620 p^{5} T^{16} + 616202 p^{6} T^{17} + 65310 p^{7} T^{18} + 5828 p^{8} T^{19} + 442 p^{9} T^{20} + 24 p^{10} T^{21} + p^{11} T^{22} \)
29 \( 1 - 10 T + 223 T^{2} - 1738 T^{3} + 23234 T^{4} - 153111 T^{5} + 1555112 T^{6} - 8967433 T^{7} + 75544650 T^{8} - 386851912 T^{9} + 2808957414 T^{10} - 12763307168 T^{11} + 2808957414 p T^{12} - 386851912 p^{2} T^{13} + 75544650 p^{3} T^{14} - 8967433 p^{4} T^{15} + 1555112 p^{5} T^{16} - 153111 p^{6} T^{17} + 23234 p^{7} T^{18} - 1738 p^{8} T^{19} + 223 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \)
31 \( 1 - 7 T + 257 T^{2} - 1224 T^{3} + 27107 T^{4} - 71823 T^{5} + 1526907 T^{6} - 266808 T^{7} + 50836506 T^{8} + 166312982 T^{9} + 1222650050 T^{10} + 8404823424 T^{11} + 1222650050 p T^{12} + 166312982 p^{2} T^{13} + 50836506 p^{3} T^{14} - 266808 p^{4} T^{15} + 1526907 p^{5} T^{16} - 71823 p^{6} T^{17} + 27107 p^{7} T^{18} - 1224 p^{8} T^{19} + 257 p^{9} T^{20} - 7 p^{10} T^{21} + p^{11} T^{22} \)
37 \( 1 + 24 T + 508 T^{2} + 7562 T^{3} + 98183 T^{4} + 1080315 T^{5} + 10633304 T^{6} + 93690019 T^{7} + 754119889 T^{8} + 5559402502 T^{9} + 37929247303 T^{10} + 239298623628 T^{11} + 37929247303 p T^{12} + 5559402502 p^{2} T^{13} + 754119889 p^{3} T^{14} + 93690019 p^{4} T^{15} + 10633304 p^{5} T^{16} + 1080315 p^{6} T^{17} + 98183 p^{7} T^{18} + 7562 p^{8} T^{19} + 508 p^{9} T^{20} + 24 p^{10} T^{21} + p^{11} T^{22} \)
41 \( 1 - 10 T + 361 T^{2} - 2954 T^{3} + 58920 T^{4} - 405370 T^{5} + 5890740 T^{6} - 34790016 T^{7} + 411606271 T^{8} - 2123444956 T^{9} + 21635231275 T^{10} - 98658081228 T^{11} + 21635231275 p T^{12} - 2123444956 p^{2} T^{13} + 411606271 p^{3} T^{14} - 34790016 p^{4} T^{15} + 5890740 p^{5} T^{16} - 405370 p^{6} T^{17} + 58920 p^{7} T^{18} - 2954 p^{8} T^{19} + 361 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \)
43 \( 1 + 14 T + 265 T^{2} + 61 p T^{3} + 30914 T^{4} + 253585 T^{5} + 2463339 T^{6} + 18078904 T^{7} + 154205784 T^{8} + 1017792937 T^{9} + 7786828379 T^{10} + 47179883218 T^{11} + 7786828379 p T^{12} + 1017792937 p^{2} T^{13} + 154205784 p^{3} T^{14} + 18078904 p^{4} T^{15} + 2463339 p^{5} T^{16} + 253585 p^{6} T^{17} + 30914 p^{7} T^{18} + 61 p^{9} T^{19} + 265 p^{9} T^{20} + 14 p^{10} T^{21} + p^{11} T^{22} \)
47 \( 1 + 8 T + 271 T^{2} + 1723 T^{3} + 34466 T^{4} + 182915 T^{5} + 2932585 T^{6} + 13925506 T^{7} + 199159662 T^{8} + 884537461 T^{9} + 11361130121 T^{10} + 46454231750 T^{11} + 11361130121 p T^{12} + 884537461 p^{2} T^{13} + 199159662 p^{3} T^{14} + 13925506 p^{4} T^{15} + 2932585 p^{5} T^{16} + 182915 p^{6} T^{17} + 34466 p^{7} T^{18} + 1723 p^{8} T^{19} + 271 p^{9} T^{20} + 8 p^{10} T^{21} + p^{11} T^{22} \)
53 \( 1 + 30 T + 774 T^{2} + 13624 T^{3} + 215507 T^{4} + 2804851 T^{5} + 33634942 T^{6} + 352803938 T^{7} + 3456326909 T^{8} + 30396300499 T^{9} + 251366508407 T^{10} + 1885206251156 T^{11} + 251366508407 p T^{12} + 30396300499 p^{2} T^{13} + 3456326909 p^{3} T^{14} + 352803938 p^{4} T^{15} + 33634942 p^{5} T^{16} + 2804851 p^{6} T^{17} + 215507 p^{7} T^{18} + 13624 p^{8} T^{19} + 774 p^{9} T^{20} + 30 p^{10} T^{21} + p^{11} T^{22} \)
59 \( 1 - 8 T + 412 T^{2} - 2481 T^{3} + 79571 T^{4} - 371950 T^{5} + 9850461 T^{6} - 36010784 T^{7} + 890530524 T^{8} - 2616920402 T^{9} + 63668204395 T^{10} - 162335636110 T^{11} + 63668204395 p T^{12} - 2616920402 p^{2} T^{13} + 890530524 p^{3} T^{14} - 36010784 p^{4} T^{15} + 9850461 p^{5} T^{16} - 371950 p^{6} T^{17} + 79571 p^{7} T^{18} - 2481 p^{8} T^{19} + 412 p^{9} T^{20} - 8 p^{10} T^{21} + p^{11} T^{22} \)
61 \( 1 + 26 T + 755 T^{2} + 13268 T^{3} + 233062 T^{4} + 3154428 T^{5} + 41724331 T^{6} + 461383941 T^{7} + 4958346070 T^{8} + 46131266978 T^{9} + 416618697741 T^{10} + 3302884114306 T^{11} + 416618697741 p T^{12} + 46131266978 p^{2} T^{13} + 4958346070 p^{3} T^{14} + 461383941 p^{4} T^{15} + 41724331 p^{5} T^{16} + 3154428 p^{6} T^{17} + 233062 p^{7} T^{18} + 13268 p^{8} T^{19} + 755 p^{9} T^{20} + 26 p^{10} T^{21} + p^{11} T^{22} \)
67 \( 1 + 24 T + 472 T^{2} + 7065 T^{3} + 97612 T^{4} + 1161790 T^{5} + 12896275 T^{6} + 130154432 T^{7} + 1259151244 T^{8} + 11346547007 T^{9} + 99181447049 T^{10} + 820140905800 T^{11} + 99181447049 p T^{12} + 11346547007 p^{2} T^{13} + 1259151244 p^{3} T^{14} + 130154432 p^{4} T^{15} + 12896275 p^{5} T^{16} + 1161790 p^{6} T^{17} + 97612 p^{7} T^{18} + 7065 p^{8} T^{19} + 472 p^{9} T^{20} + 24 p^{10} T^{21} + p^{11} T^{22} \)
71 \( 1 - 16 T + 554 T^{2} - 6896 T^{3} + 145110 T^{4} - 1534334 T^{5} + 24610471 T^{6} - 226711103 T^{7} + 2999784508 T^{8} - 24338836255 T^{9} + 276344642609 T^{10} - 1976888122261 T^{11} + 276344642609 p T^{12} - 24338836255 p^{2} T^{13} + 2999784508 p^{3} T^{14} - 226711103 p^{4} T^{15} + 24610471 p^{5} T^{16} - 1534334 p^{6} T^{17} + 145110 p^{7} T^{18} - 6896 p^{8} T^{19} + 554 p^{9} T^{20} - 16 p^{10} T^{21} + p^{11} T^{22} \)
79 \( 1 - 4 T + 476 T^{2} - 1176 T^{3} + 110161 T^{4} - 105022 T^{5} + 16456340 T^{6} + 9395908 T^{7} + 1816050591 T^{8} + 3278308706 T^{9} + 163836709073 T^{10} + 365273462376 T^{11} + 163836709073 p T^{12} + 3278308706 p^{2} T^{13} + 1816050591 p^{3} T^{14} + 9395908 p^{4} T^{15} + 16456340 p^{5} T^{16} - 105022 p^{6} T^{17} + 110161 p^{7} T^{18} - 1176 p^{8} T^{19} + 476 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \)
83 \( 1 + 2 T + 461 T^{2} + 1015 T^{3} + 112608 T^{4} + 179977 T^{5} + 18631435 T^{6} + 17424017 T^{7} + 2312804318 T^{8} + 979924729 T^{9} + 230973195772 T^{10} + 50302998510 T^{11} + 230973195772 p T^{12} + 979924729 p^{2} T^{13} + 2312804318 p^{3} T^{14} + 17424017 p^{4} T^{15} + 18631435 p^{5} T^{16} + 179977 p^{6} T^{17} + 112608 p^{7} T^{18} + 1015 p^{8} T^{19} + 461 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \)
89 \( 1 + 11 T + 502 T^{2} + 6196 T^{3} + 141801 T^{4} + 1695873 T^{5} + 27999374 T^{6} + 306289652 T^{7} + 4128733239 T^{8} + 40457344893 T^{9} + 470570998181 T^{10} + 4085868483855 T^{11} + 470570998181 p T^{12} + 40457344893 p^{2} T^{13} + 4128733239 p^{3} T^{14} + 306289652 p^{4} T^{15} + 27999374 p^{5} T^{16} + 1695873 p^{6} T^{17} + 141801 p^{7} T^{18} + 6196 p^{8} T^{19} + 502 p^{9} T^{20} + 11 p^{10} T^{21} + p^{11} T^{22} \)
97 \( 1 + 37 T + 1259 T^{2} + 28648 T^{3} + 599870 T^{4} + 10335813 T^{5} + 165344750 T^{6} + 2327278047 T^{7} + 30573118178 T^{8} + 363365195332 T^{9} + 4042305666926 T^{10} + 41131327564942 T^{11} + 4042305666926 p T^{12} + 363365195332 p^{2} T^{13} + 30573118178 p^{3} T^{14} + 2327278047 p^{4} T^{15} + 165344750 p^{5} T^{16} + 10335813 p^{6} T^{17} + 599870 p^{7} T^{18} + 28648 p^{8} T^{19} + 1259 p^{9} T^{20} + 37 p^{10} T^{21} + p^{11} T^{22} \)
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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

−2.81173493334253370353279342739, −2.73695364698541065055155838674, −2.67335769197217497933735983887, −2.33259498637042909934262826420, −2.29935814926219356198393769938, −2.20313654639485393913410490166, −2.17694997159191777898703323553, −2.08789789956527561438535517042, −2.03866324968351553276028319288, −2.01466793690463925487440876805, −1.95159273437106382143558188586, −1.92948897212113603693013777650, −1.89928768547939558191672618720, −1.71904479760588982615582021039, −1.68539876282212026844502650662, −1.44901740124774815247512097148, −1.43812417456871051370451042862, −1.32360938118406360512800371257, −1.30027092173428305035559780557, −1.13548464259047681965481674227, −1.11615568650225888869257932308, −1.08251667589429837202559324109, −1.06438947683115022971970174620, −0.975254778130705960411851470579, −0.895516997988804701707758955031, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.895516997988804701707758955031, 0.975254778130705960411851470579, 1.06438947683115022971970174620, 1.08251667589429837202559324109, 1.11615568650225888869257932308, 1.13548464259047681965481674227, 1.30027092173428305035559780557, 1.32360938118406360512800371257, 1.43812417456871051370451042862, 1.44901740124774815247512097148, 1.68539876282212026844502650662, 1.71904479760588982615582021039, 1.89928768547939558191672618720, 1.92948897212113603693013777650, 1.95159273437106382143558188586, 2.01466793690463925487440876805, 2.03866324968351553276028319288, 2.08789789956527561438535517042, 2.17694997159191777898703323553, 2.20313654639485393913410490166, 2.29935814926219356198393769938, 2.33259498637042909934262826420, 2.67335769197217497933735983887, 2.73695364698541065055155838674, 2.81173493334253370353279342739

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

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