Properties

Label 24-637e12-1.1-c1e12-0-8
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  − 4·2-s − 3-s − 5-s + 4·6-s + 22·8-s + 11·9-s + 4·10-s + 4·11-s + 2·13-s + 15-s − 32·16-s + 10·17-s − 44·18-s + 19-s − 16·22-s + 2·23-s − 22·24-s + 19·25-s − 8·26-s − 10·27-s + 3·29-s − 4·30-s − 16·31-s − 20·32-s − 4·33-s − 40·34-s + 26·37-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.577·3-s − 0.447·5-s + 1.63·6-s + 7.77·8-s + 11/3·9-s + 1.26·10-s + 1.20·11-s + 0.554·13-s + 0.258·15-s − 8·16-s + 2.42·17-s − 10.3·18-s + 0.229·19-s − 3.41·22-s + 0.417·23-s − 4.49·24-s + 19/5·25-s − 1.56·26-s − 1.92·27-s + 0.557·29-s − 0.730·30-s − 2.87·31-s − 3.53·32-s − 0.696·33-s − 6.85·34-s + 4.27·37-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

\(L(1)\) \(\approx\) \(0.4417567660\)
\(L(\frac12)\) \(\approx\) \(0.4417567660\)
\(L(\frac{3}{2})\) 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 \( ( 1 + p T + 3 p T^{2} + 9 T^{3} + 5 p^{2} T^{4} + 7 p^{2} T^{5} + 51 T^{6} + 7 p^{3} T^{7} + 5 p^{4} T^{8} + 9 p^{3} T^{9} + 3 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \)
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 \( 1 + T - 18 T^{2} + p T^{3} + 193 T^{4} - 192 T^{5} - 1181 T^{6} + 2139 T^{7} + 908 p T^{8} - 451 p^{2} T^{9} - 6679 T^{10} + 26266 T^{11} + 249 T^{12} + 26266 p T^{13} - 6679 p^{2} T^{14} - 451 p^{5} T^{15} + 908 p^{5} T^{16} + 2139 p^{5} T^{17} - 1181 p^{6} T^{18} - 192 p^{7} T^{19} + 193 p^{8} T^{20} + p^{10} T^{21} - 18 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
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 \( ( 1 - 5 T + 90 T^{2} - 411 T^{3} + 3539 T^{4} - 13744 T^{5} + 78123 T^{6} - 13744 p T^{7} + 3539 p^{2} T^{8} - 411 p^{3} T^{9} + 90 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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 \( ( 1 - T + 32 T^{2} - 52 T^{3} + 1214 T^{4} - 1383 T^{5} + 21935 T^{6} - 1383 p T^{7} + 1214 p^{2} T^{8} - 52 p^{3} T^{9} + 32 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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 \( 1 + 16 T + 20 T^{2} - 594 T^{3} + 2163 T^{4} + 43649 T^{5} - 56125 T^{6} - 1282696 T^{7} + 2984747 T^{8} + 22743273 T^{9} - 180497697 T^{10} - 655302586 T^{11} + 2182678017 T^{12} - 655302586 p T^{13} - 180497697 p^{2} T^{14} + 22743273 p^{3} T^{15} + 2984747 p^{4} T^{16} - 1282696 p^{5} T^{17} - 56125 p^{6} T^{18} + 43649 p^{7} T^{19} + 2163 p^{8} T^{20} - 594 p^{9} T^{21} + 20 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \)
37 \( ( 1 - 13 T + 184 T^{2} - 1054 T^{3} + 7158 T^{4} - 10573 T^{5} + 113729 T^{6} - 10573 p T^{7} + 7158 p^{2} T^{8} - 1054 p^{3} T^{9} + 184 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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 \( 1 - T - 104 T^{2} + 189 T^{3} + 5335 T^{4} - 164 p T^{5} - 69863 T^{6} - 514255 T^{7} - 7627520 T^{8} + 55687467 T^{9} + 662939941 T^{10} - 1686387922 T^{11} - 35399065407 T^{12} - 1686387922 p T^{13} + 662939941 p^{2} T^{14} + 55687467 p^{3} T^{15} - 7627520 p^{4} T^{16} - 514255 p^{5} T^{17} - 69863 p^{6} T^{18} - 164 p^{8} T^{19} + 5335 p^{8} T^{20} + 189 p^{9} T^{21} - 104 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 + 2 T - 214 T^{2} - 252 T^{3} + 24796 T^{4} + 13772 T^{5} - 1921862 T^{6} + 82142 T^{7} + 113089342 T^{8} - 43114584 T^{9} - 5653831794 T^{10} + 1443208718 T^{11} + 285781391787 T^{12} + 1443208718 p T^{13} - 5653831794 p^{2} T^{14} - 43114584 p^{3} T^{15} + 113089342 p^{4} T^{16} + 82142 p^{5} T^{17} - 1921862 p^{6} T^{18} + 13772 p^{7} T^{19} + 24796 p^{8} T^{20} - 252 p^{9} T^{21} - 214 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
59 \( ( 1 - 13 T + 5 p T^{2} - 2839 T^{3} + 38957 T^{4} - 294699 T^{5} + 2963017 T^{6} - 294699 p T^{7} + 38957 p^{2} T^{8} - 2839 p^{3} T^{9} + 5 p^{5} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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 \( 1 - 30 T + 224 T^{2} + 1118 T^{3} - 5021 T^{4} - 290169 T^{5} + 1854677 T^{6} + 9817892 T^{7} - 27971653 T^{8} - 688598777 T^{9} - 1819010273 T^{10} + 20701972840 T^{11} + 235631264151 T^{12} + 20701972840 p T^{13} - 1819010273 p^{2} T^{14} - 688598777 p^{3} T^{15} - 27971653 p^{4} T^{16} + 9817892 p^{5} T^{17} + 1854677 p^{6} T^{18} - 290169 p^{7} T^{19} - 5021 p^{8} T^{20} + 1118 p^{9} T^{21} + 224 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 7 T - 277 T^{2} + 2628 T^{3} + 34995 T^{4} - 387429 T^{5} - 3070086 T^{6} + 26237658 T^{7} + 339376855 T^{8} - 565746882 T^{9} - 45365142063 T^{10} - 8895648284 T^{11} + 4474615429807 T^{12} - 8895648284 p T^{13} - 45365142063 p^{2} T^{14} - 565746882 p^{3} T^{15} + 339376855 p^{4} T^{16} + 26237658 p^{5} T^{17} - 3070086 p^{6} T^{18} - 387429 p^{7} T^{19} + 34995 p^{8} T^{20} + 2628 p^{9} T^{21} - 277 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \)
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 \( ( 1 - 4 T + 167 T^{2} - 1648 T^{3} + 21035 T^{4} - 202110 T^{5} + 2204075 T^{6} - 202110 p T^{7} + 21035 p^{2} T^{8} - 1648 p^{3} T^{9} + 167 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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

−3.52683143997779211880704356583, −3.50589051999421539977626693469, −3.45383237494748981207863400597, −3.10619359074108145553187176376, −3.00386571948342543341490619828, −2.92934278500421617752703630122, −2.87563965479997819645837348140, −2.83753304758176781872754146496, −2.52479101997313739219425835019, −2.28768504223823744189439885375, −2.28492842216554993704442385648, −2.19552020756986417624906489263, −2.18762863127060956220944978898, −1.93368373237992002653196236689, −1.66812778624682861465806820869, −1.51937051905495275059087021785, −1.35379884565889016145538646438, −1.34070592054714039825533346688, −1.13233025484344767286346958763, −1.06037455747608530093793971085, −0.909927814578637257170161400004, −0.811997872974530704723468565031, −0.67884128888582850562125857081, −0.57832258904084789940288647068, −0.19818429921661631477491650500, 0.19818429921661631477491650500, 0.57832258904084789940288647068, 0.67884128888582850562125857081, 0.811997872974530704723468565031, 0.909927814578637257170161400004, 1.06037455747608530093793971085, 1.13233025484344767286346958763, 1.34070592054714039825533346688, 1.35379884565889016145538646438, 1.51937051905495275059087021785, 1.66812778624682861465806820869, 1.93368373237992002653196236689, 2.18762863127060956220944978898, 2.19552020756986417624906489263, 2.28492842216554993704442385648, 2.28768504223823744189439885375, 2.52479101997313739219425835019, 2.83753304758176781872754146496, 2.87563965479997819645837348140, 2.92934278500421617752703630122, 3.00386571948342543341490619828, 3.10619359074108145553187176376, 3.45383237494748981207863400597, 3.50589051999421539977626693469, 3.52683143997779211880704356583

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

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