Properties

Label 24-325e12-1.1-c1e12-0-1
Degree $24$
Conductor $1.389\times 10^{30}$
Sign $1$
Analytic cond. $93309.4$
Root an. cond. $1.61094$
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  + 4·4-s + 6·9-s + 6·16-s − 12·19-s − 18·29-s − 16·31-s + 24·36-s + 14·41-s + 18·49-s + 4·59-s + 6·61-s − 6·64-s − 12·71-s − 48·76-s + 104·79-s + 26·81-s − 20·89-s − 26·101-s − 24·109-s − 72·116-s + 40·121-s − 64·124-s + 127-s + 131-s + 137-s + 139-s + 36·144-s + ⋯
L(s)  = 1  + 2·4-s + 2·9-s + 3/2·16-s − 2.75·19-s − 3.34·29-s − 2.87·31-s + 4·36-s + 2.18·41-s + 18/7·49-s + 0.520·59-s + 0.768·61-s − 3/4·64-s − 1.42·71-s − 5.50·76-s + 11.7·79-s + 26/9·81-s − 2.11·89-s − 2.58·101-s − 2.29·109-s − 6.68·116-s + 3.63·121-s − 5.74·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(5^{24} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(93309.4\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{325} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 5^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.712969930\)
\(L(\frac12)\) \(\approx\) \(5.712969930\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + 15 T^{2} + 3 p T^{4} + 322 T^{6} + 3 p^{3} T^{8} + 15 p^{4} T^{10} + p^{6} T^{12} \)
good2 \( 1 - p^{2} T^{2} + 5 p T^{4} - 5 p T^{6} - p^{4} T^{8} + 43 p T^{10} - 223 T^{12} + 43 p^{3} T^{14} - p^{8} T^{16} - 5 p^{7} T^{18} + 5 p^{9} T^{20} - p^{12} T^{22} + p^{12} T^{24} \)
3 \( 1 - 2 p T^{2} + 10 T^{4} + 40 T^{6} - 146 T^{8} - 226 T^{10} + 2062 T^{12} - 226 p^{2} T^{14} - 146 p^{4} T^{16} + 40 p^{6} T^{18} + 10 p^{8} T^{20} - 2 p^{11} T^{22} + p^{12} T^{24} \)
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

−3.83417313808108094677518552446, −3.82628270283402294002025052495, −3.75029450958973146217343882193, −3.73443216338418249785763348930, −3.57770421711716431169448419990, −3.49488531013592124361709342929, −3.32441812126338217459835954609, −3.31558468809429596259582130457, −3.06054334795429395109780906596, −2.82794462003446795043598963213, −2.51296369904954113382691381509, −2.50954520661204715337587000227, −2.45051787942386466549826812519, −2.30937594473853540571085422387, −2.29052616586120195996422410584, −2.18553822503122477012390471528, −2.07716826168265005742498599522, −1.99429700966625358077612533408, −1.64041750570228439160641096303, −1.54487802309859865535166810401, −1.47798420532240754271086828667, −1.31237083377814264592273014348, −0.965832160718243140185441821246, −0.63602583674095764609159264387, −0.35153418649211656866614430062, 0.35153418649211656866614430062, 0.63602583674095764609159264387, 0.965832160718243140185441821246, 1.31237083377814264592273014348, 1.47798420532240754271086828667, 1.54487802309859865535166810401, 1.64041750570228439160641096303, 1.99429700966625358077612533408, 2.07716826168265005742498599522, 2.18553822503122477012390471528, 2.29052616586120195996422410584, 2.30937594473853540571085422387, 2.45051787942386466549826812519, 2.50954520661204715337587000227, 2.51296369904954113382691381509, 2.82794462003446795043598963213, 3.06054334795429395109780906596, 3.31558468809429596259582130457, 3.32441812126338217459835954609, 3.49488531013592124361709342929, 3.57770421711716431169448419990, 3.73443216338418249785763348930, 3.75029450958973146217343882193, 3.82628270283402294002025052495, 3.83417313808108094677518552446

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

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