Properties

Label 24-1040e12-1.1-c1e12-0-4
Degree $24$
Conductor $1.601\times 10^{36}$
Sign $1$
Analytic cond. $1.07578\times 10^{11}$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 6·9-s − 12·19-s + 17·25-s + 18·29-s + 16·31-s + 14·41-s + 36·45-s − 18·49-s + 4·59-s + 6·61-s + 12·71-s + 104·79-s + 26·81-s + 20·89-s + 72·95-s − 26·101-s + 24·109-s + 40·121-s − 38·125-s + 127-s + 131-s + 137-s + 139-s − 108·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2.68·5-s − 2·9-s − 2.75·19-s + 17/5·25-s + 3.34·29-s + 2.87·31-s + 2.18·41-s + 5.36·45-s − 2.57·49-s + 0.520·59-s + 0.768·61-s + 1.42·71-s + 11.7·79-s + 26/9·81-s + 2.11·89-s + 7.38·95-s − 2.58·101-s + 2.29·109-s + 3.63·121-s − 3.39·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8.96·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 5^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(1.07578\times 10^{11}\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1040} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 5^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.205426368\)
\(L(\frac12)\) \(\approx\) \(6.205426368\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + 3 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 15 T^{2} + 3 p T^{4} - 322 T^{6} + 3 p^{3} T^{8} - 15 p^{4} T^{10} + p^{6} T^{12} \)
good3 \( 1 + 2 p T^{2} + 10 T^{4} - 40 T^{6} - 146 T^{8} + 226 T^{10} + 2062 T^{12} + 226 p^{2} T^{14} - 146 p^{4} T^{16} - 40 p^{6} T^{18} + 10 p^{8} T^{20} + 2 p^{11} T^{22} + p^{12} T^{24} \)
7 \( 1 + 18 T^{2} + 82 T^{4} + 536 T^{6} + 12326 T^{8} + 78550 T^{10} + 235838 T^{12} + 78550 p^{2} T^{14} + 12326 p^{4} T^{16} + 536 p^{6} T^{18} + 82 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \)
11 \( ( 1 - 20 T^{2} + 16 T^{3} + 180 T^{4} - 160 T^{5} - 1674 T^{6} - 160 p T^{7} + 180 p^{2} T^{8} + 16 p^{3} T^{9} - 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 67 T^{2} + 2371 T^{4} + 56020 T^{6} + 996641 T^{8} + 14527513 T^{10} + 221018582 T^{12} + 14527513 p^{2} T^{14} + 996641 p^{4} T^{16} + 56020 p^{6} T^{18} + 2371 p^{8} T^{20} + 67 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 + 6 T - 20 T^{2} - 100 T^{3} + 764 T^{4} + 1606 T^{5} - 11338 T^{6} + 1606 p T^{7} + 764 p^{2} T^{8} - 100 p^{3} T^{9} - 20 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 126 T^{2} + 9010 T^{4} + 451400 T^{6} + 17445734 T^{8} + 540254266 T^{10} + 13705453502 T^{12} + 540254266 p^{2} T^{14} + 17445734 p^{4} T^{16} + 451400 p^{6} T^{18} + 9010 p^{8} T^{20} + 126 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{6} \)
31 \( ( 1 - 4 T + 53 T^{2} - 288 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
37 \( 1 + 187 T^{2} + 19451 T^{4} + 1426940 T^{6} + 81658601 T^{8} + 3849109833 T^{10} + 153778997622 T^{12} + 3849109833 p^{2} T^{14} + 81658601 p^{4} T^{16} + 1426940 p^{6} T^{18} + 19451 p^{8} T^{20} + 187 p^{10} T^{22} + p^{12} T^{24} \)
41 \( ( 1 - 7 T - 45 T^{2} + 500 T^{3} + 929 T^{4} - 12237 T^{5} + 28438 T^{6} - 12237 p T^{7} + 929 p^{2} T^{8} + 500 p^{3} T^{9} - 45 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
43 \( 1 + 178 T^{2} + 17426 T^{4} + 1084520 T^{6} + 47574566 T^{8} + 1584620742 T^{10} + 56532979182 T^{12} + 1584620742 p^{2} T^{14} + 47574566 p^{4} T^{16} + 1084520 p^{6} T^{18} + 17426 p^{8} T^{20} + 178 p^{10} T^{22} + p^{12} T^{24} \)
47 \( ( 1 - 46 T^{2} + 3407 T^{4} - 54276 T^{6} + 3407 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 147 T^{2} + 6923 T^{4} - 205346 T^{6} + 6923 p^{2} T^{8} - 147 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 - 2 T - 2 p T^{2} - 44 T^{3} + 7486 T^{4} + 9630 T^{5} - 471322 T^{6} + 9630 p T^{7} + 7486 p^{2} T^{8} - 44 p^{3} T^{9} - 2 p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 3 T - 125 T^{2} + 100 T^{3} + 9029 T^{4} + 4247 T^{5} - 611842 T^{6} + 4247 p T^{7} + 9029 p^{2} T^{8} + 100 p^{3} T^{9} - 125 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 302 T^{2} + 48066 T^{4} + 5726680 T^{6} + 564790166 T^{8} + 46598346138 T^{10} + 3315667632142 T^{12} + 46598346138 p^{2} T^{14} + 564790166 p^{4} T^{16} + 5726680 p^{6} T^{18} + 48066 p^{8} T^{20} + 302 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 - 6 T - 176 T^{2} + 380 T^{3} + 24936 T^{4} - 24734 T^{5} - 1974402 T^{6} - 24734 p T^{7} + 24936 p^{2} T^{8} + 380 p^{3} T^{9} - 176 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 223 T^{2} + 31055 T^{4} - 2685330 T^{6} + 31055 p^{2} T^{8} - 223 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 26 T + 417 T^{2} - 4268 T^{3} + 417 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( ( 1 - 222 T^{2} + 35303 T^{4} - 3305156 T^{6} + 35303 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 10 T - 10 T^{2} - 8 p T^{3} + 370 T^{4} + 95250 T^{5} - 421426 T^{6} + 95250 p T^{7} + 370 p^{2} T^{8} - 8 p^{4} T^{9} - 10 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( 1 + 302 T^{2} + 44346 T^{4} + 4094200 T^{6} + 2434958 p T^{8} - 3936373062 T^{10} - 1781970067778 T^{12} - 3936373062 p^{2} T^{14} + 2434958 p^{5} T^{16} + 4094200 p^{6} T^{18} + 44346 p^{8} T^{20} + 302 p^{10} T^{22} + p^{12} T^{24} \)
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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

−3.27466293185922342458617306613, −3.07140790453103644085971398829, −2.93047262022320151326568229657, −2.90078242788430391758831434452, −2.86767596741634844516185830015, −2.61548845689197633152467966488, −2.60095601349640785186624101007, −2.50926166875816843744106645503, −2.46044794595188180361529456942, −2.36643465732124748367853660498, −2.21744156692565254759346125640, −2.02359898693882865543271878335, −1.99107244110755776837276699043, −1.86579009227951594763340139866, −1.77303967059882865097425424468, −1.65670644237685268914681205620, −1.59190360034883034003677036905, −1.11414172672724402846744736124, −0.900655008674030558555509546236, −0.861264723354494521898680042382, −0.792197199837449202727660201934, −0.790275786605588814677851577688, −0.50867349451479540649118703690, −0.38092368172820071579314715862, −0.35760653168332151520447911181, 0.35760653168332151520447911181, 0.38092368172820071579314715862, 0.50867349451479540649118703690, 0.790275786605588814677851577688, 0.792197199837449202727660201934, 0.861264723354494521898680042382, 0.900655008674030558555509546236, 1.11414172672724402846744736124, 1.59190360034883034003677036905, 1.65670644237685268914681205620, 1.77303967059882865097425424468, 1.86579009227951594763340139866, 1.99107244110755776837276699043, 2.02359898693882865543271878335, 2.21744156692565254759346125640, 2.36643465732124748367853660498, 2.46044794595188180361529456942, 2.50926166875816843744106645503, 2.60095601349640785186624101007, 2.61548845689197633152467966488, 2.86767596741634844516185830015, 2.90078242788430391758831434452, 2.93047262022320151326568229657, 3.07140790453103644085971398829, 3.27466293185922342458617306613

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

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