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Properties

Label 24-91e24-1.1-c1e12-0-4
Degree $24$
Conductor $1.040\times 10^{47}$
Sign $1$
Analytic cond. $6.98744\times 10^{21}$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $12$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 4·4-s + 34·8-s − 4·9-s − 20·11-s + 4·16-s + 16·18-s + 80·22-s − 10·25-s + 2·29-s − 168·32-s + 16·36-s − 10·37-s + 28·43-s + 80·44-s + 40·50-s − 30·53-s − 8·58-s + 21·64-s + 32·67-s − 86·71-s − 136·72-s + 40·74-s + 28·79-s − 11·81-s − 112·86-s − 680·88-s + ⋯
L(s)  = 1  − 2.82·2-s − 2·4-s + 12.0·8-s − 4/3·9-s − 6.03·11-s + 16-s + 3.77·18-s + 17.0·22-s − 2·25-s + 0.371·29-s − 29.6·32-s + 8/3·36-s − 1.64·37-s + 4.26·43-s + 12.0·44-s + 5.65·50-s − 4.12·53-s − 1.05·58-s + 21/8·64-s + 3.90·67-s − 10.2·71-s − 16.0·72-s + 4.64·74-s + 3.15·79-s − 1.22·81-s − 12.0·86-s − 72.4·88-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(7^{24} \cdot 13^{24}\)
Sign: $1$
Analytic conductor: \(6.98744\times 10^{21}\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(12\)
Selberg data: \((24,\ 7^{24} \cdot 13^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

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)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( ( 1 + p T + p^{3} T^{2} + 15 T^{3} + p^{5} T^{4} + 13 p^{2} T^{5} + 79 T^{6} + 13 p^{3} T^{7} + p^{7} T^{8} + 15 p^{3} T^{9} + p^{7} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \)
3 \( 1 + 4 T^{2} + p^{3} T^{4} + 26 p T^{6} + 394 T^{8} + 377 p T^{10} + 4555 T^{12} + 377 p^{3} T^{14} + 394 p^{4} T^{16} + 26 p^{7} T^{18} + p^{11} T^{20} + 4 p^{10} T^{22} + p^{12} T^{24} \)
5 \( 1 + 2 p T^{2} + 106 T^{4} + 793 T^{6} + 5053 T^{8} + 30373 T^{10} + 152617 T^{12} + 30373 p^{2} T^{14} + 5053 p^{4} T^{16} + 793 p^{6} T^{18} + 106 p^{8} T^{20} + 2 p^{11} T^{22} + p^{12} T^{24} \)
11 \( ( 1 + 10 T + 8 p T^{2} + 505 T^{3} + 2619 T^{4} + 10565 T^{5} + 39013 T^{6} + 10565 p T^{7} + 2619 p^{2} T^{8} + 505 p^{3} T^{9} + 8 p^{5} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
17 \( 1 + 25 T^{2} + 622 T^{4} + 11992 T^{6} + 249314 T^{8} + 4870303 T^{10} + 100826423 T^{12} + 4870303 p^{2} T^{14} + 249314 p^{4} T^{16} + 11992 p^{6} T^{18} + 622 p^{8} T^{20} + 25 p^{10} T^{22} + p^{12} T^{24} \)
19 \( 1 + 63 T^{2} + 2215 T^{4} + 54159 T^{6} + 1078891 T^{8} + 18700143 T^{10} + 338044993 T^{12} + 18700143 p^{2} T^{14} + 1078891 p^{4} T^{16} + 54159 p^{6} T^{18} + 2215 p^{8} T^{20} + 63 p^{10} T^{22} + p^{12} T^{24} \)
23 \( ( 1 + 71 T^{2} - 147 T^{3} + 2677 T^{4} - 6482 T^{5} + 75915 T^{6} - 6482 p T^{7} + 2677 p^{2} T^{8} - 147 p^{3} T^{9} + 71 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - T + 89 T^{2} - 161 T^{3} + 4787 T^{4} - 7769 T^{5} + 167919 T^{6} - 7769 p T^{7} + 4787 p^{2} T^{8} - 161 p^{3} T^{9} + 89 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( 1 + 99 T^{2} + 6589 T^{4} + 316413 T^{6} + 13312945 T^{8} + 467436717 T^{10} + 15327031561 T^{12} + 467436717 p^{2} T^{14} + 13312945 p^{4} T^{16} + 316413 p^{6} T^{18} + 6589 p^{8} T^{20} + 99 p^{10} T^{22} + p^{12} T^{24} \)
37 \( ( 1 + 5 T + 187 T^{2} + 885 T^{3} + 15621 T^{4} + 63925 T^{5} + 745015 T^{6} + 63925 p T^{7} + 15621 p^{2} T^{8} + 885 p^{3} T^{9} + 187 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 + 290 T^{2} + 42053 T^{4} + 4069582 T^{6} + 293632802 T^{8} + 16633697082 T^{10} + 757765751493 T^{12} + 16633697082 p^{2} T^{14} + 293632802 p^{4} T^{16} + 4069582 p^{6} T^{18} + 42053 p^{8} T^{20} + 290 p^{10} T^{22} + p^{12} T^{24} \)
43 \( ( 1 - 14 T + 285 T^{2} - 2786 T^{3} + 31866 T^{4} - 230517 T^{5} + 1846261 T^{6} - 230517 p T^{7} + 31866 p^{2} T^{8} - 2786 p^{3} T^{9} + 285 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( 1 + 196 T^{2} + 18202 T^{4} + 1087219 T^{6} + 58743809 T^{8} + 3460976323 T^{10} + 184492693973 T^{12} + 3460976323 p^{2} T^{14} + 58743809 p^{4} T^{16} + 1087219 p^{6} T^{18} + 18202 p^{8} T^{20} + 196 p^{10} T^{22} + p^{12} T^{24} \)
53 \( ( 1 + 15 T + 283 T^{2} + 2685 T^{3} + 31040 T^{4} + 4238 p T^{5} + 2013191 T^{6} + 4238 p^{2} T^{7} + 31040 p^{2} T^{8} + 2685 p^{3} T^{9} + 283 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
59 \( 1 + 358 T^{2} + 66141 T^{4} + 8390971 T^{6} + 812028543 T^{8} + 63309251140 T^{10} + 4088454622565 T^{12} + 63309251140 p^{2} T^{14} + 812028543 p^{4} T^{16} + 8390971 p^{6} T^{18} + 66141 p^{8} T^{20} + 358 p^{10} T^{22} + p^{12} T^{24} \)
61 \( 1 + 356 T^{2} + 64591 T^{4} + 7999690 T^{6} + 757637442 T^{8} + 58647515095 T^{10} + 3857915526551 T^{12} + 58647515095 p^{2} T^{14} + 757637442 p^{4} T^{16} + 7999690 p^{6} T^{18} + 64591 p^{8} T^{20} + 356 p^{10} T^{22} + p^{12} T^{24} \)
67 \( ( 1 - 16 T + 306 T^{2} - 2976 T^{3} + 38655 T^{4} - 319047 T^{5} + 3301261 T^{6} - 319047 p T^{7} + 38655 p^{2} T^{8} - 2976 p^{3} T^{9} + 306 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 + 43 T + 1134 T^{2} + 20775 T^{3} + 296232 T^{4} + 3361771 T^{5} + 31328851 T^{6} + 3361771 p T^{7} + 296232 p^{2} T^{8} + 20775 p^{3} T^{9} + 1134 p^{4} T^{10} + 43 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( 1 + 397 T^{2} + 73858 T^{4} + 9199000 T^{6} + 944937970 T^{8} + 86973276163 T^{10} + 6932987633959 T^{12} + 86973276163 p^{2} T^{14} + 944937970 p^{4} T^{16} + 9199000 p^{6} T^{18} + 73858 p^{8} T^{20} + 397 p^{10} T^{22} + p^{12} T^{24} \)
79 \( ( 1 - 14 T + 281 T^{2} - 2471 T^{3} + 37509 T^{4} - 309820 T^{5} + 3846549 T^{6} - 309820 p T^{7} + 37509 p^{2} T^{8} - 2471 p^{3} T^{9} + 281 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 + 199 T^{2} + 31200 T^{4} + 4039735 T^{6} + 456148608 T^{8} + 45962720059 T^{10} + 3917366196869 T^{12} + 45962720059 p^{2} T^{14} + 456148608 p^{4} T^{16} + 4039735 p^{6} T^{18} + 31200 p^{8} T^{20} + 199 p^{10} T^{22} + p^{12} T^{24} \)
89 \( 1 + 382 T^{2} + 81992 T^{4} + 12821758 T^{6} + 1602460467 T^{8} + 170799999069 T^{10} + 16119873258779 T^{12} + 170799999069 p^{2} T^{14} + 1602460467 p^{4} T^{16} + 12821758 p^{6} T^{18} + 81992 p^{8} T^{20} + 382 p^{10} T^{22} + p^{12} T^{24} \)
97 \( 1 + 823 T^{2} + 328586 T^{4} + 872455 p T^{6} + 15696715833 T^{8} + 2209813634136 T^{10} + 242088550457549 T^{12} + 2209813634136 p^{2} T^{14} + 15696715833 p^{4} T^{16} + 872455 p^{7} T^{18} + 328586 p^{8} T^{20} + 823 p^{10} T^{22} + p^{12} T^{24} \)
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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

−2.64768956804687601262079672241, −2.61208185192208560855377029826, −2.55217096653423673118926564152, −2.53082999892019246863840555940, −2.48887669296083913941057645570, −2.40804682175903259759315755331, −2.28952479679896630561016396973, −2.24970489459190887579373686341, −2.22964439868859592037314752123, −1.97812362169874249251083693124, −1.93714482985956192287575222545, −1.88379477936388402263314846930, −1.70711212413766758332282846393, −1.67862782384248741714698815672, −1.47201350480708957772392994282, −1.43906991734018726875603363609, −1.25943366805606206798481675081, −1.14599718073866420463125677825, −1.11431887999327158942341978616, −1.06439766706438559351820675173, −1.04089232863436913789016315734, −0.966364076102494155320040490938, −0.938887354181080851611398729392, −0.848312089534830010518779349618, −0.76008403556558240258301880067, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.76008403556558240258301880067, 0.848312089534830010518779349618, 0.938887354181080851611398729392, 0.966364076102494155320040490938, 1.04089232863436913789016315734, 1.06439766706438559351820675173, 1.11431887999327158942341978616, 1.14599718073866420463125677825, 1.25943366805606206798481675081, 1.43906991734018726875603363609, 1.47201350480708957772392994282, 1.67862782384248741714698815672, 1.70711212413766758332282846393, 1.88379477936388402263314846930, 1.93714482985956192287575222545, 1.97812362169874249251083693124, 2.22964439868859592037314752123, 2.24970489459190887579373686341, 2.28952479679896630561016396973, 2.40804682175903259759315755331, 2.48887669296083913941057645570, 2.53082999892019246863840555940, 2.55217096653423673118926564152, 2.61208185192208560855377029826, 2.64768956804687601262079672241

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

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