Properties

Label 24-1323e12-1.1-c1e12-0-1
Degree $24$
Conductor $2.876\times 10^{37}$
Sign $1$
Analytic cond. $1.93216\times 10^{12}$
Root an. cond. $3.25026$
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  + 2·2-s + 5·4-s + 2·8-s − 16·11-s + 2·16-s − 32·22-s − 8·23-s − 18·25-s + 22·29-s − 12·32-s + 6·37-s − 6·43-s − 80·44-s − 16·46-s − 36·50-s + 28·53-s + 44·58-s − 17·64-s − 76·71-s + 12·74-s + 6·79-s − 12·86-s − 32·88-s − 40·92-s − 90·100-s + 56·106-s + 26·107-s + ⋯
L(s)  = 1  + 1.41·2-s + 5/2·4-s + 0.707·8-s − 4.82·11-s + 1/2·16-s − 6.82·22-s − 1.66·23-s − 3.59·25-s + 4.08·29-s − 2.12·32-s + 0.986·37-s − 0.914·43-s − 12.0·44-s − 2.35·46-s − 5.09·50-s + 3.84·53-s + 5.77·58-s − 2.12·64-s − 9.01·71-s + 1.39·74-s + 0.675·79-s − 1.29·86-s − 3.41·88-s − 4.17·92-s − 9·100-s + 5.43·106-s + 2.51·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{36} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(1.93216\times 10^{12}\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{36} \cdot 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7154401959\)
\(L(\frac12)\) \(\approx\) \(0.7154401959\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 - T - T^{2} + p^{2} T^{3} - 3 T^{4} - p T^{5} + 13 T^{6} - p^{2} T^{7} - 3 p^{2} T^{8} + p^{5} T^{9} - p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \)
5 \( ( 1 + 9 T^{2} + 63 T^{4} + 349 T^{6} + 63 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 + 4 T + 32 T^{2} + 87 T^{3} + 32 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
13 \( 1 - 3 p T^{2} + 51 p T^{4} - 6584 T^{6} + 5157 p T^{8} - 102645 p T^{10} + 22407342 T^{12} - 102645 p^{3} T^{14} + 5157 p^{5} T^{16} - 6584 p^{6} T^{18} + 51 p^{9} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24} \)
17 \( 1 - 18 T^{2} + 108 T^{4} + 1706 T^{6} - 66114 T^{8} + 139734 T^{10} + 15553959 T^{12} + 139734 p^{2} T^{14} - 66114 p^{4} T^{16} + 1706 p^{6} T^{18} + 108 p^{8} T^{20} - 18 p^{10} T^{22} + p^{12} T^{24} \)
19 \( 1 - 39 T^{2} + 510 T^{4} + 31 p T^{6} - 123201 T^{8} + 3156120 T^{10} - 72752079 T^{12} + 3156120 p^{2} T^{14} - 123201 p^{4} T^{16} + 31 p^{7} T^{18} + 510 p^{8} T^{20} - 39 p^{10} T^{22} + p^{12} T^{24} \)
23 \( ( 1 + 2 T + 44 T^{2} + 33 T^{3} + 44 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
29 \( ( 1 - 11 T + 20 T^{2} - 13 T^{3} + 1233 T^{4} - 262 T^{5} - 47411 T^{6} - 262 p T^{7} + 1233 p^{2} T^{8} - 13 p^{3} T^{9} + 20 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( 1 - 57 T^{2} - 678 T^{4} + 7483 T^{6} + 5054805 T^{8} - 2471706 p T^{10} - 2776215 p^{2} T^{12} - 2471706 p^{3} T^{14} + 5054805 p^{4} T^{16} + 7483 p^{6} T^{18} - 678 p^{8} T^{20} - 57 p^{10} T^{22} + p^{12} T^{24} \)
37 \( ( 1 - 3 T - 78 T^{2} + 237 T^{3} + 3603 T^{4} - 6456 T^{5} - 129067 T^{6} - 6456 p T^{7} + 3603 p^{2} T^{8} + 237 p^{3} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 - 84 T^{2} + 2334 T^{4} + 75506 T^{6} - 5470866 T^{8} - 18588942 T^{10} + 7812254391 T^{12} - 18588942 p^{2} T^{14} - 5470866 p^{4} T^{16} + 75506 p^{6} T^{18} + 2334 p^{8} T^{20} - 84 p^{10} T^{22} + p^{12} T^{24} \)
43 \( ( 1 + 3 T - 96 T^{2} - 255 T^{3} + 5655 T^{4} + 8382 T^{5} - 250477 T^{6} + 8382 p T^{7} + 5655 p^{2} T^{8} - 255 p^{3} T^{9} - 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( 1 - 99 T^{2} + 486 T^{4} - 19399 T^{6} + 21504483 T^{8} - 613600884 T^{10} - 14184900399 T^{12} - 613600884 p^{2} T^{14} + 21504483 p^{4} T^{16} - 19399 p^{6} T^{18} + 486 p^{8} T^{20} - 99 p^{10} T^{22} + p^{12} T^{24} \)
53 \( ( 1 - 14 T + 26 T^{2} + 62 T^{3} + 2796 T^{4} + 5384 T^{5} - 293669 T^{6} + 5384 p T^{7} + 2796 p^{2} T^{8} + 62 p^{3} T^{9} + 26 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
59 \( 1 - 171 T^{2} + 10764 T^{4} - 658021 T^{6} + 63198909 T^{8} - 3185695998 T^{10} + 106058651361 T^{12} - 3185695998 p^{2} T^{14} + 63198909 p^{4} T^{16} - 658021 p^{6} T^{18} + 10764 p^{8} T^{20} - 171 p^{10} T^{22} + p^{12} T^{24} \)
61 \( 1 - 102 T^{2} - 2514 T^{4} + 108094 T^{6} + 46094616 T^{8} - 815218740 T^{10} - 153775720821 T^{12} - 815218740 p^{2} T^{14} + 46094616 p^{4} T^{16} + 108094 p^{6} T^{18} - 2514 p^{8} T^{20} - 102 p^{10} T^{22} + p^{12} T^{24} \)
67 \( ( 1 - 90 T^{2} + 706 T^{3} + 2070 T^{4} - 31770 T^{5} + 183435 T^{6} - 31770 p T^{7} + 2070 p^{2} T^{8} + 706 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( ( 1 + 19 T + 329 T^{2} + 2925 T^{3} + 329 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
73 \( 1 - 363 T^{2} + 72438 T^{4} - 10402079 T^{6} + 1182976731 T^{8} - 111093122556 T^{10} + 8799155948049 T^{12} - 111093122556 p^{2} T^{14} + 1182976731 p^{4} T^{16} - 10402079 p^{6} T^{18} + 72438 p^{8} T^{20} - 363 p^{10} T^{22} + p^{12} T^{24} \)
79 \( ( 1 - 3 T - 150 T^{2} + 257 T^{3} + 11619 T^{4} - 1710 T^{5} - 932601 T^{6} - 1710 p T^{7} + 11619 p^{2} T^{8} + 257 p^{3} T^{9} - 150 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 - 270 T^{2} + 31896 T^{4} - 3065782 T^{6} + 318136230 T^{8} - 24401026026 T^{10} + 1632699460815 T^{12} - 24401026026 p^{2} T^{14} + 318136230 p^{4} T^{16} - 3065782 p^{6} T^{18} + 31896 p^{8} T^{20} - 270 p^{10} T^{22} + p^{12} T^{24} \)
89 \( 1 - 288 T^{2} + 45954 T^{4} - 3797734 T^{6} + 88086726 T^{8} + 23906597562 T^{10} - 3299038805433 T^{12} + 23906597562 p^{2} T^{14} + 88086726 p^{4} T^{16} - 3797734 p^{6} T^{18} + 45954 p^{8} T^{20} - 288 p^{10} T^{22} + p^{12} T^{24} \)
97 \( 1 - 471 T^{2} + 121776 T^{4} - 22400861 T^{6} + 3247635573 T^{8} - 392084795286 T^{10} + 40704346255641 T^{12} - 392084795286 p^{2} T^{14} + 3247635573 p^{4} T^{16} - 22400861 p^{6} T^{18} + 121776 p^{8} T^{20} - 471 p^{10} T^{22} + p^{12} T^{24} \)
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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

−2.89942236397684887495704802629, −2.85733809442905448687527767904, −2.83920383859375930781129306478, −2.80495227076343575108738673590, −2.76250764390182906548691438075, −2.62100368993071932562510263900, −2.57913835387543878902828257225, −2.34497216771989950401843735928, −2.31801222719081076095649801449, −2.30482839566329170636242778910, −2.28474505584235212381072481845, −2.13399412754712448695229563478, −1.91545955109300635071564356572, −1.79423938124565297113334380195, −1.69568617851227494755837155947, −1.45638642423796687837297759581, −1.44252647681626216748853835202, −1.41034628489346599728112932949, −1.16588928774620046349954382398, −1.12736692980086614084263645505, −0.870409223148638665232873813034, −0.52646847791738160531861880026, −0.32566413189893949728223362840, −0.24694211979477312987432548151, −0.099133898907884163371417469189, 0.099133898907884163371417469189, 0.24694211979477312987432548151, 0.32566413189893949728223362840, 0.52646847791738160531861880026, 0.870409223148638665232873813034, 1.12736692980086614084263645505, 1.16588928774620046349954382398, 1.41034628489346599728112932949, 1.44252647681626216748853835202, 1.45638642423796687837297759581, 1.69568617851227494755837155947, 1.79423938124565297113334380195, 1.91545955109300635071564356572, 2.13399412754712448695229563478, 2.28474505584235212381072481845, 2.30482839566329170636242778910, 2.31801222719081076095649801449, 2.34497216771989950401843735928, 2.57913835387543878902828257225, 2.62100368993071932562510263900, 2.76250764390182906548691438075, 2.80495227076343575108738673590, 2.83920383859375930781129306478, 2.85733809442905448687527767904, 2.89942236397684887495704802629

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

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