Properties

Label 24-637e12-1.1-c1e12-0-13
Degree $24$
Conductor $4.463\times 10^{33}$
Sign $1$
Analytic cond. $2.99915\times 10^{8}$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 6·4-s + 2·5-s − 2·6-s + 10·8-s + 11·9-s + 4·10-s + 4·11-s − 6·12-s + 2·13-s − 2·15-s + 22·16-s − 5·17-s + 22·18-s + 19-s + 12·20-s + 8·22-s − 23-s − 10·24-s − 35·25-s + 4·26-s − 10·27-s + 3·29-s − 4·30-s + 32·31-s + 34·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3·4-s + 0.894·5-s − 0.816·6-s + 3.53·8-s + 11/3·9-s + 1.26·10-s + 1.20·11-s − 1.73·12-s + 0.554·13-s − 0.516·15-s + 11/2·16-s − 1.21·17-s + 5.18·18-s + 0.229·19-s + 2.68·20-s + 1.70·22-s − 0.208·23-s − 2.04·24-s − 7·25-s + 0.784·26-s − 1.92·27-s + 0.557·29-s − 0.730·30-s + 5.74·31-s + 6.01·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(7^{24} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(2.99915\times 10^{8}\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 7^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(53.45256869\)
\(L(\frac12)\) \(\approx\) \(53.45256869\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 - 2 T - 16 T^{2} - 3 T^{3} + 607 T^{4} - 433 T^{5} - 5615 T^{6} - 433 p T^{7} + 607 p^{2} T^{8} - 3 p^{3} T^{9} - 16 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
good2 \( 1 - p T - p T^{2} + 3 p T^{3} - p T^{4} - p T^{5} + 7 T^{6} - p T^{7} - 7 p T^{8} + 3 p T^{9} - 5 p^{2} T^{10} - 5 p^{2} T^{11} + 153 T^{12} - 5 p^{3} T^{13} - 5 p^{4} T^{14} + 3 p^{4} T^{15} - 7 p^{5} T^{16} - p^{6} T^{17} + 7 p^{6} T^{18} - p^{8} T^{19} - p^{9} T^{20} + 3 p^{10} T^{21} - p^{11} T^{22} - p^{12} T^{23} + p^{12} T^{24} \)
3 \( 1 + T - 10 T^{2} - 11 T^{3} + 53 T^{4} + 62 T^{5} - 167 T^{6} - 221 T^{7} + 98 p T^{8} + 535 T^{9} + 79 T^{10} - 604 T^{11} - 1559 T^{12} - 604 p T^{13} + 79 p^{2} T^{14} + 535 p^{3} T^{15} + 98 p^{5} T^{16} - 221 p^{5} T^{17} - 167 p^{6} T^{18} + 62 p^{7} T^{19} + 53 p^{8} T^{20} - 11 p^{9} T^{21} - 10 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
5 \( ( 1 - T + 19 T^{2} - 7 T^{3} + 161 T^{4} + 3 T^{5} + 913 T^{6} + 3 p T^{7} + 161 p^{2} T^{8} - 7 p^{3} T^{9} + 19 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \)
11 \( 1 - 4 T - 29 T^{2} + 108 T^{3} + 477 T^{4} - 113 p T^{5} - 6686 T^{6} + 7665 T^{7} + 89323 T^{8} + 423 T^{9} - 1282040 T^{10} - 249219 T^{11} + 16505087 T^{12} - 249219 p T^{13} - 1282040 p^{2} T^{14} + 423 p^{3} T^{15} + 89323 p^{4} T^{16} + 7665 p^{5} T^{17} - 6686 p^{6} T^{18} - 113 p^{8} T^{19} + 477 p^{8} T^{20} + 108 p^{9} T^{21} - 29 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 + 5 T - 65 T^{2} - 372 T^{3} + 2506 T^{4} + 15344 T^{5} - 62063 T^{6} - 395128 T^{7} + 1158376 T^{8} + 6526599 T^{9} - 17123414 T^{10} - 46016896 T^{11} + 268434807 T^{12} - 46016896 p T^{13} - 17123414 p^{2} T^{14} + 6526599 p^{3} T^{15} + 1158376 p^{4} T^{16} - 395128 p^{5} T^{17} - 62063 p^{6} T^{18} + 15344 p^{7} T^{19} + 2506 p^{8} T^{20} - 372 p^{9} T^{21} - 65 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 - T - 49 T^{2} - 82 T^{3} + 1336 T^{4} + 4335 T^{5} - 10907 T^{6} - 99626 T^{7} - 263580 T^{8} + 1110690 T^{9} + 9684539 T^{10} - 2414194 T^{11} - 215227743 T^{12} - 2414194 p T^{13} + 9684539 p^{2} T^{14} + 1110690 p^{3} T^{15} - 263580 p^{4} T^{16} - 99626 p^{5} T^{17} - 10907 p^{6} T^{18} + 4335 p^{7} T^{19} + 1336 p^{8} T^{20} - 82 p^{9} T^{21} - 49 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + T - 31 T^{2} - 72 T^{3} - 242 T^{4} + 619 T^{5} + 6343 T^{6} + 28360 T^{7} + 121372 T^{8} - 9024 p T^{9} + 5143119 T^{10} - 4729028 T^{11} - 273608319 T^{12} - 4729028 p T^{13} + 5143119 p^{2} T^{14} - 9024 p^{4} T^{15} + 121372 p^{4} T^{16} + 28360 p^{5} T^{17} + 6343 p^{6} T^{18} + 619 p^{7} T^{19} - 242 p^{8} T^{20} - 72 p^{9} T^{21} - 31 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 3 T - 3 p T^{2} + 94 T^{3} + 158 p T^{4} + 904 T^{5} - 123467 T^{6} - 308042 T^{7} + 1215550 T^{8} + 10832851 T^{9} + 73693658 T^{10} - 174959016 T^{11} - 3324017493 T^{12} - 174959016 p T^{13} + 73693658 p^{2} T^{14} + 10832851 p^{3} T^{15} + 1215550 p^{4} T^{16} - 308042 p^{5} T^{17} - 123467 p^{6} T^{18} + 904 p^{7} T^{19} + 158 p^{9} T^{20} + 94 p^{9} T^{21} - 3 p^{11} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
31 \( ( 1 - 16 T + 236 T^{2} - 2185 T^{3} + 18573 T^{4} - 122325 T^{5} + 755039 T^{6} - 122325 p T^{7} + 18573 p^{2} T^{8} - 2185 p^{3} T^{9} + 236 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( 1 + 13 T - 15 T^{2} + 284 T^{3} + 12996 T^{4} + 18401 T^{5} - 116147 T^{6} + 5523346 T^{7} + 19538810 T^{8} - 71463812 T^{9} + 1452640399 T^{10} + 7689412934 T^{11} - 18842100883 T^{12} + 7689412934 p T^{13} + 1452640399 p^{2} T^{14} - 71463812 p^{3} T^{15} + 19538810 p^{4} T^{16} + 5523346 p^{5} T^{17} - 116147 p^{6} T^{18} + 18401 p^{7} T^{19} + 12996 p^{8} T^{20} + 284 p^{9} T^{21} - 15 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 - 8 T - 161 T^{2} + 924 T^{3} + 18241 T^{4} - 64367 T^{5} - 1502654 T^{6} + 3175261 T^{7} + 96068491 T^{8} - 105301221 T^{9} - 5078164754 T^{10} + 1647875431 T^{11} + 226350132753 T^{12} + 1647875431 p T^{13} - 5078164754 p^{2} T^{14} - 105301221 p^{3} T^{15} + 96068491 p^{4} T^{16} + 3175261 p^{5} T^{17} - 1502654 p^{6} T^{18} - 64367 p^{7} T^{19} + 18241 p^{8} T^{20} + 924 p^{9} T^{21} - 161 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 + 11 T - 138 T^{2} - 1349 T^{3} + 16370 T^{4} + 106653 T^{5} - 1472431 T^{6} - 5757651 T^{7} + 106708219 T^{8} + 224797058 T^{9} - 6088028976 T^{10} - 3777766292 T^{11} + 288640495545 T^{12} - 3777766292 p T^{13} - 6088028976 p^{2} T^{14} + 224797058 p^{3} T^{15} + 106708219 p^{4} T^{16} - 5757651 p^{5} T^{17} - 1472431 p^{6} T^{18} + 106653 p^{7} T^{19} + 16370 p^{8} T^{20} - 1349 p^{9} T^{21} - 138 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
47 \( ( 1 + T + 105 T^{2} + 147 T^{3} + 5543 T^{4} + 3359 T^{5} + 246951 T^{6} + 3359 p T^{7} + 5543 p^{2} T^{8} + 147 p^{3} T^{9} + 105 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 2 T + 218 T^{2} - 344 T^{3} + 22040 T^{4} - 26940 T^{5} + 1409201 T^{6} - 26940 p T^{7} + 22040 p^{2} T^{8} - 344 p^{3} T^{9} + 218 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
59 \( 1 + 13 T - 126 T^{2} - 1843 T^{3} + 11161 T^{4} + 119322 T^{5} - 1337447 T^{6} - 7367025 T^{7} + 123366322 T^{8} + 382593671 T^{9} - 9090177085 T^{10} - 8156701016 T^{11} + 592237594305 T^{12} - 8156701016 p T^{13} - 9090177085 p^{2} T^{14} + 382593671 p^{3} T^{15} + 123366322 p^{4} T^{16} - 7367025 p^{5} T^{17} - 1337447 p^{6} T^{18} + 119322 p^{7} T^{19} + 11161 p^{8} T^{20} - 1843 p^{9} T^{21} - 126 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 - 5 T - 140 T^{2} + 373 T^{3} + 8487 T^{4} + 5202 T^{5} - 147441 T^{6} - 963135 T^{7} - 4711566 T^{8} + 13690661 T^{9} - 1296684385 T^{10} + 689962304 T^{11} + 162150963097 T^{12} + 689962304 p T^{13} - 1296684385 p^{2} T^{14} + 13690661 p^{3} T^{15} - 4711566 p^{4} T^{16} - 963135 p^{5} T^{17} - 147441 p^{6} T^{18} + 5202 p^{7} T^{19} + 8487 p^{8} T^{20} + 373 p^{9} T^{21} - 140 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 11 T - 175 T^{2} - 2336 T^{3} + 15663 T^{4} + 247450 T^{5} - 15954 p T^{6} - 18125445 T^{7} + 60512732 T^{8} + 977936543 T^{9} - 2490157221 T^{10} - 26393757979 T^{11} + 95373451231 T^{12} - 26393757979 p T^{13} - 2490157221 p^{2} T^{14} + 977936543 p^{3} T^{15} + 60512732 p^{4} T^{16} - 18125445 p^{5} T^{17} - 15954 p^{7} T^{18} + 247450 p^{7} T^{19} + 15663 p^{8} T^{20} - 2336 p^{9} T^{21} - 175 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 - 6 T - 249 T^{2} + 278 T^{3} + 39793 T^{4} + 68141 T^{5} - 3761552 T^{6} - 15648583 T^{7} + 241594531 T^{8} + 1275513473 T^{9} - 10122739162 T^{10} - 44683203723 T^{11} + 523547364015 T^{12} - 44683203723 p T^{13} - 10122739162 p^{2} T^{14} + 1275513473 p^{3} T^{15} + 241594531 p^{4} T^{16} - 15648583 p^{5} T^{17} - 3761552 p^{6} T^{18} + 68141 p^{7} T^{19} + 39793 p^{8} T^{20} + 278 p^{9} T^{21} - 249 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
73 \( ( 1 + 30 T + 676 T^{2} + 10699 T^{3} + 141027 T^{4} + 1519265 T^{5} + 14149139 T^{6} + 1519265 p T^{7} + 141027 p^{2} T^{8} + 10699 p^{3} T^{9} + 676 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 + 7 T + 326 T^{2} + 2455 T^{3} + 54096 T^{4} + 344443 T^{5} + 5474643 T^{6} + 344443 p T^{7} + 54096 p^{2} T^{8} + 2455 p^{3} T^{9} + 326 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 27 T + 656 T^{2} - 10802 T^{3} + 153994 T^{4} - 1760871 T^{5} + 17670883 T^{6} - 1760871 p T^{7} + 153994 p^{2} T^{8} - 10802 p^{3} T^{9} + 656 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
89 \( 1 + 4 T - 151 T^{2} - 2628 T^{3} + 262 T^{4} + 309046 T^{5} + 2802769 T^{6} - 6034970 T^{7} - 281402495 T^{8} - 1666391304 T^{9} + 5367237150 T^{10} + 95837476354 T^{11} + 701675320941 T^{12} + 95837476354 p T^{13} + 5367237150 p^{2} T^{14} - 1666391304 p^{3} T^{15} - 281402495 p^{4} T^{16} - 6034970 p^{5} T^{17} + 2802769 p^{6} T^{18} + 309046 p^{7} T^{19} + 262 p^{8} T^{20} - 2628 p^{9} T^{21} - 151 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 - 35 T + 278 T^{2} + 3177 T^{3} - 20496 T^{4} - 1111333 T^{5} + 13328183 T^{6} + 54713297 T^{7} - 1182920923 T^{8} - 11775176076 T^{9} + 251445486222 T^{10} - 186060844192 T^{11} - 17274836413101 T^{12} - 186060844192 p T^{13} + 251445486222 p^{2} T^{14} - 11775176076 p^{3} T^{15} - 1182920923 p^{4} T^{16} + 54713297 p^{5} T^{17} + 13328183 p^{6} T^{18} - 1111333 p^{7} T^{19} - 20496 p^{8} T^{20} + 3177 p^{9} T^{21} + 278 p^{10} T^{22} - 35 p^{11} T^{23} + p^{12} T^{24} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.52613901911939569504501647789, −3.52435549322508809895105736344, −3.43087224757493627613410966375, −3.07634512198213374880369624647, −3.01331195750254048415041859487, −2.94296140772031489149230943743, −2.87624789559039379976685943527, −2.82920376722463379904109306164, −2.56424057468339963261823210863, −2.50399705313134471980680001288, −2.39125458713730210177750496910, −2.08986965524769999493118566556, −1.98402621044789217797760315432, −1.97773268463008196544616204740, −1.95517915759830916865595483719, −1.95350651225808498191316490983, −1.76103362439323387243986848632, −1.54195222339506745114845247166, −1.46713675594188325938413675365, −1.41917180428381505523179430213, −1.15551818511442488990243480011, −1.01612972670841535045981335641, −0.949522021807354355401760335051, −0.54480117693191687495316566520, −0.31277695431341298370700645824, 0.31277695431341298370700645824, 0.54480117693191687495316566520, 0.949522021807354355401760335051, 1.01612972670841535045981335641, 1.15551818511442488990243480011, 1.41917180428381505523179430213, 1.46713675594188325938413675365, 1.54195222339506745114845247166, 1.76103362439323387243986848632, 1.95350651225808498191316490983, 1.95517915759830916865595483719, 1.97773268463008196544616204740, 1.98402621044789217797760315432, 2.08986965524769999493118566556, 2.39125458713730210177750496910, 2.50399705313134471980680001288, 2.56424057468339963261823210863, 2.82920376722463379904109306164, 2.87624789559039379976685943527, 2.94296140772031489149230943743, 3.01331195750254048415041859487, 3.07634512198213374880369624647, 3.43087224757493627613410966375, 3.52435549322508809895105736344, 3.52613901911939569504501647789

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

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