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Properties

Label 24-845e12-1.1-c1e12-0-0
Degree $24$
Conductor $1.325\times 10^{35}$
Sign $1$
Analytic cond. $8.90442\times 10^{9}$
Root an. cond. $2.59756$
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  − 6·2-s + 19·4-s + 4·5-s + 4·7-s − 38·8-s − 6·9-s − 24·10-s − 24·14-s + 51·16-s + 36·18-s + 76·20-s + 6·25-s + 76·28-s + 12·29-s − 40·32-s + 16·35-s − 114·36-s − 152·40-s − 24·45-s − 40·47-s + 38·49-s − 36·50-s − 152·56-s − 72·58-s − 12·61-s − 24·63-s + 14·64-s + ⋯
L(s)  = 1  − 4.24·2-s + 19/2·4-s + 1.78·5-s + 1.51·7-s − 13.4·8-s − 2·9-s − 7.58·10-s − 6.41·14-s + 51/4·16-s + 8.48·18-s + 16.9·20-s + 6/5·25-s + 14.3·28-s + 2.22·29-s − 7.07·32-s + 2.70·35-s − 19·36-s − 24.0·40-s − 3.57·45-s − 5.83·47-s + 38/7·49-s − 5.09·50-s − 20.3·56-s − 9.45·58-s − 1.53·61-s − 3.02·63-s + 7/4·64-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(5^{12} \cdot 13^{24}\)
Sign: $1$
Analytic conductor: \(8.90442\times 10^{9}\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 5^{12} \cdot 13^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.008368945122\)
\(L(\frac12)\) \(\approx\) \(0.008368945122\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 \)
good2 \( ( 1 + 3 T + p^{2} T^{2} + T^{3} - 3 p T^{4} - 13 T^{5} - 21 T^{6} - 13 p T^{7} - 3 p^{3} T^{8} + p^{3} T^{9} + p^{6} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
3 \( 1 + 2 p T^{2} + 25 T^{4} + 50 T^{6} - 26 T^{8} - 614 T^{10} - 2483 T^{12} - 614 p^{2} T^{14} - 26 p^{4} T^{16} + 50 p^{6} T^{18} + 25 p^{8} T^{20} + 2 p^{11} T^{22} + p^{12} T^{24} \)
7 \( ( 1 - 2 T - 13 T^{2} + 2 p T^{3} + 122 T^{4} - 30 T^{5} - 957 T^{6} - 30 p T^{7} + 122 p^{2} T^{8} + 2 p^{4} T^{9} - 13 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
11 \( 1 + 46 T^{2} + 1093 T^{4} + 18994 T^{6} + 277274 T^{8} + 330926 p T^{10} + 42829781 T^{12} + 330926 p^{3} T^{14} + 277274 p^{4} T^{16} + 18994 p^{6} T^{18} + 1093 p^{8} T^{20} + 46 p^{10} T^{22} + p^{12} T^{24} \)
17 \( 1 + 58 T^{2} + 1909 T^{4} + 32974 T^{6} + 205466 T^{8} - 6528062 T^{10} - 183608467 T^{12} - 6528062 p^{2} T^{14} + 205466 p^{4} T^{16} + 32974 p^{6} T^{18} + 1909 p^{8} T^{20} + 58 p^{10} T^{22} + p^{12} T^{24} \)
19 \( 1 + 106 T^{2} + 6413 T^{4} + 270326 T^{6} + 8751890 T^{8} + 225659478 T^{10} + 4742113173 T^{12} + 225659478 p^{2} T^{14} + 8751890 p^{4} T^{16} + 270326 p^{6} T^{18} + 6413 p^{8} T^{20} + 106 p^{10} T^{22} + p^{12} T^{24} \)
23 \( 1 + 66 T^{2} + 1609 T^{4} + 34358 T^{6} + 1243694 T^{8} + 28479574 T^{10} + 483380405 T^{12} + 28479574 p^{2} T^{14} + 1243694 p^{4} T^{16} + 34358 p^{6} T^{18} + 1609 p^{8} T^{20} + 66 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 - 6 T - 15 T^{2} + 6 p T^{3} - 318 T^{4} + 462 T^{5} + 673 T^{6} + 462 p T^{7} - 318 p^{2} T^{8} + 6 p^{4} T^{9} - 15 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 126 T^{2} + 7895 T^{4} - 306224 T^{6} + 7895 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 83 T^{2} - 104 T^{3} + 3818 T^{4} + 4316 T^{5} - 150883 T^{6} + 4316 p T^{7} + 3818 p^{2} T^{8} - 104 p^{3} T^{9} - 83 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( 1 + 166 T^{2} + 14693 T^{4} + 868274 T^{6} + 38674394 T^{8} + 1438946526 T^{10} + 54812823741 T^{12} + 1438946526 p^{2} T^{14} + 38674394 p^{4} T^{16} + 868274 p^{6} T^{18} + 14693 p^{8} T^{20} + 166 p^{10} T^{22} + p^{12} T^{24} \)
43 \( 1 + 130 T^{2} + 5729 T^{4} + 328838 T^{6} + 31115582 T^{8} + 1348803606 T^{10} + 39628381005 T^{12} + 1348803606 p^{2} T^{14} + 31115582 p^{4} T^{16} + 328838 p^{6} T^{18} + 5729 p^{8} T^{20} + 130 p^{10} T^{22} + p^{12} T^{24} \)
47 \( ( 1 + 10 T + 169 T^{2} + 960 T^{3} + 169 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
53 \( ( 1 - 174 T^{2} + 18071 T^{4} - 1143332 T^{6} + 18071 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( 1 + 210 T^{2} + 20077 T^{4} + 1615214 T^{6} + 133370786 T^{8} + 8617371934 T^{10} + 481651058597 T^{12} + 8617371934 p^{2} T^{14} + 133370786 p^{4} T^{16} + 1615214 p^{6} T^{18} + 20077 p^{8} T^{20} + 210 p^{10} T^{22} + p^{12} T^{24} \)
61 \( ( 1 + 6 T - 131 T^{2} - 470 T^{3} + 13286 T^{4} + 22994 T^{5} - 839527 T^{6} + 22994 p T^{7} + 13286 p^{2} T^{8} - 470 p^{3} T^{9} - 131 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( ( 1 + 10 T - 41 T^{2} - 62 T^{3} + 3074 T^{4} - 40274 T^{5} - 637705 T^{6} - 40274 p T^{7} + 3074 p^{2} T^{8} - 62 p^{3} T^{9} - 41 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( 1 + 106 T^{2} + 661 T^{4} - 39578 T^{6} - 2985934 T^{8} - 2471487914 T^{10} - 256356950803 T^{12} - 2471487914 p^{2} T^{14} - 2985934 p^{4} T^{16} - 39578 p^{6} T^{18} + 661 p^{8} T^{20} + 106 p^{10} T^{22} + p^{12} T^{24} \)
73 \( ( 1 + 24 T + 383 T^{2} + 3740 T^{3} + 383 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
79 \( ( 1 - 16 T + 261 T^{2} - 2512 T^{3} + 261 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( ( 1 + 22 T + 401 T^{2} + 3968 T^{3} + 401 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
89 \( 1 + 330 T^{2} + 56005 T^{4} + 6340574 T^{6} + 544961690 T^{8} + 38135323090 T^{10} + 2883878320541 T^{12} + 38135323090 p^{2} T^{14} + 544961690 p^{4} T^{16} + 6340574 p^{6} T^{18} + 56005 p^{8} T^{20} + 330 p^{10} T^{22} + p^{12} T^{24} \)
97 \( ( 1 - 14 T - 11 T^{2} + 22 p T^{3} - 12454 T^{4} - 90326 T^{5} + 2155085 T^{6} - 90326 p T^{7} - 12454 p^{2} T^{8} + 22 p^{4} T^{9} - 11 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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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

−3.13916229045468365309214688801, −3.06923125174859647602160561262, −2.98926848746696963183225079686, −2.88966255126029878626930188203, −2.82233496253710463910651867823, −2.78497393676324224979229724358, −2.63625443274584448577786710557, −2.31152212972794523392284221340, −2.29664765204884823590316885037, −2.20740439666095851770032035167, −2.20043268194000132913341704866, −2.16932909325876718810700479512, −1.96055626101953945790542241032, −1.72911563668004761609646242691, −1.63702951593784523780096215103, −1.47698453695986166873603540821, −1.39249809660299041489897962010, −1.34218711657373796781727985802, −1.25367066038681728563285530051, −1.19451284276438749688684187128, −1.02781284350167616449447770322, −1.00331680381374674875877548940, −0.51016715159850527574386368823, −0.19450798317359261294366957117, −0.03584555687581315868030790527, 0.03584555687581315868030790527, 0.19450798317359261294366957117, 0.51016715159850527574386368823, 1.00331680381374674875877548940, 1.02781284350167616449447770322, 1.19451284276438749688684187128, 1.25367066038681728563285530051, 1.34218711657373796781727985802, 1.39249809660299041489897962010, 1.47698453695986166873603540821, 1.63702951593784523780096215103, 1.72911563668004761609646242691, 1.96055626101953945790542241032, 2.16932909325876718810700479512, 2.20043268194000132913341704866, 2.20740439666095851770032035167, 2.29664765204884823590316885037, 2.31152212972794523392284221340, 2.63625443274584448577786710557, 2.78497393676324224979229724358, 2.82233496253710463910651867823, 2.88966255126029878626930188203, 2.98926848746696963183225079686, 3.06923125174859647602160561262, 3.13916229045468365309214688801

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

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