Properties

Label 24-4560e12-1.1-c1e12-0-1
Degree $24$
Conductor $8.083\times 10^{43}$
Sign $1$
Analytic cond. $5.43129\times 10^{18}$
Root an. cond. $6.03421$
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  − 12·3-s + 12·5-s + 78·9-s − 144·15-s + 4·17-s + 78·25-s − 364·27-s + 12·31-s + 936·45-s + 28·49-s − 48·51-s − 52·59-s − 56·61-s + 32·67-s − 8·71-s + 32·73-s − 936·75-s + 28·79-s + 1.36e3·81-s + 48·85-s − 144·93-s + 24·101-s − 32·103-s − 56·107-s + 64·121-s + 364·125-s + 127-s + ⋯
L(s)  = 1  − 6.92·3-s + 5.36·5-s + 26·9-s − 37.1·15-s + 0.970·17-s + 78/5·25-s − 70.0·27-s + 2.15·31-s + 139.·45-s + 4·49-s − 6.72·51-s − 6.76·59-s − 7.17·61-s + 3.90·67-s − 0.949·71-s + 3.74·73-s − 108.·75-s + 3.15·79-s + 151.·81-s + 5.20·85-s − 14.9·93-s + 2.38·101-s − 3.15·103-s − 5.41·107-s + 5.81·121-s + 32.5·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(5.43129\times 10^{18}\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 3^{12} \cdot 5^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.328706340\)
\(L(\frac12)\) \(\approx\) \(3.328706340\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T )^{12} \)
5 \( ( 1 - T )^{12} \)
19 \( 1 + 26 T^{2} + 64 T^{3} + 471 T^{4} + 2368 T^{5} + 4652 T^{6} + 2368 p T^{7} + 471 p^{2} T^{8} + 64 p^{3} T^{9} + 26 p^{4} T^{10} + p^{6} T^{12} \)
good7 \( 1 - 4 p T^{2} + 66 p T^{4} - 5788 T^{6} + 58659 T^{8} - 506888 T^{10} + 3808092 T^{12} - 506888 p^{2} T^{14} + 58659 p^{4} T^{16} - 5788 p^{6} T^{18} + 66 p^{9} T^{20} - 4 p^{11} T^{22} + p^{12} T^{24} \)
11 \( 1 - 64 T^{2} + 194 p T^{4} - 4464 p T^{6} + 876627 T^{8} - 12769216 T^{10} + 154124140 T^{12} - 12769216 p^{2} T^{14} + 876627 p^{4} T^{16} - 4464 p^{7} T^{18} + 194 p^{9} T^{20} - 64 p^{10} T^{22} + p^{12} T^{24} \)
13 \( 1 - 40 T^{2} + 966 T^{4} - 14056 T^{6} + 142227 T^{8} - 855872 T^{10} + 6120396 T^{12} - 855872 p^{2} T^{14} + 142227 p^{4} T^{16} - 14056 p^{6} T^{18} + 966 p^{8} T^{20} - 40 p^{10} T^{22} + p^{12} T^{24} \)
17 \( ( 1 - 2 T + 38 T^{2} - 66 T^{3} + 803 T^{4} - 980 T^{5} + 12572 T^{6} - 980 p T^{7} + 803 p^{2} T^{8} - 66 p^{3} T^{9} + 38 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 - 76 T^{2} + 3658 T^{4} - 129564 T^{6} + 3585471 T^{8} - 3794216 p T^{10} + 2016083884 T^{12} - 3794216 p^{3} T^{14} + 3585471 p^{4} T^{16} - 129564 p^{6} T^{18} + 3658 p^{8} T^{20} - 76 p^{10} T^{22} + p^{12} T^{24} \)
29 \( 1 - 8 p T^{2} + 878 p T^{4} - 1777704 T^{6} + 89791731 T^{8} - 3532351936 T^{10} + 112869552172 T^{12} - 3532351936 p^{2} T^{14} + 89791731 p^{4} T^{16} - 1777704 p^{6} T^{18} + 878 p^{9} T^{20} - 8 p^{11} T^{22} + p^{12} T^{24} \)
31 \( ( 1 - 6 T + 134 T^{2} - 674 T^{3} + 8691 T^{4} - 36020 T^{5} + 337484 T^{6} - 36020 p T^{7} + 8691 p^{2} T^{8} - 674 p^{3} T^{9} + 134 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( 1 - 88 T^{2} + 6966 T^{4} - 424840 T^{6} + 22510035 T^{8} - 1008193712 T^{10} + 39632581164 T^{12} - 1008193712 p^{2} T^{14} + 22510035 p^{4} T^{16} - 424840 p^{6} T^{18} + 6966 p^{8} T^{20} - 88 p^{10} T^{22} + p^{12} T^{24} \)
41 \( 1 - 284 T^{2} + 41118 T^{4} - 3993452 T^{6} + 289087587 T^{8} - 16408240408 T^{10} + 748005955452 T^{12} - 16408240408 p^{2} T^{14} + 289087587 p^{4} T^{16} - 3993452 p^{6} T^{18} + 41118 p^{8} T^{20} - 284 p^{10} T^{22} + p^{12} T^{24} \)
43 \( 1 - 112 T^{2} + 10902 T^{4} - 706960 T^{6} + 39797523 T^{8} - 1952687312 T^{10} + 85963943532 T^{12} - 1952687312 p^{2} T^{14} + 39797523 p^{4} T^{16} - 706960 p^{6} T^{18} + 10902 p^{8} T^{20} - 112 p^{10} T^{22} + p^{12} T^{24} \)
47 \( 1 - 380 T^{2} + 69930 T^{4} - 8342156 T^{6} + 724428447 T^{8} - 48406756792 T^{10} + 2552197195116 T^{12} - 48406756792 p^{2} T^{14} + 724428447 p^{4} T^{16} - 8342156 p^{6} T^{18} + 69930 p^{8} T^{20} - 380 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 - 364 T^{2} + 68218 T^{4} - 8607516 T^{6} + 810449967 T^{8} - 59748149848 T^{10} + 3527294199052 T^{12} - 59748149848 p^{2} T^{14} + 810449967 p^{4} T^{16} - 8607516 p^{6} T^{18} + 68218 p^{8} T^{20} - 364 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 + 26 T + 426 T^{2} + 5110 T^{3} + 52087 T^{4} + 451748 T^{5} + 3628076 T^{6} + 451748 p T^{7} + 52087 p^{2} T^{8} + 5110 p^{3} T^{9} + 426 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 28 T + 534 T^{2} + 7468 T^{3} + 86523 T^{4} + 13736 p T^{5} + 7078476 T^{6} + 13736 p^{2} T^{7} + 86523 p^{2} T^{8} + 7468 p^{3} T^{9} + 534 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 16 T + 282 T^{2} - 2920 T^{3} + 33663 T^{4} - 303512 T^{5} + 2786268 T^{6} - 303512 p T^{7} + 33663 p^{2} T^{8} - 2920 p^{3} T^{9} + 282 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 + 4 T + 330 T^{2} + 980 T^{3} + 49975 T^{4} + 117328 T^{5} + 4494428 T^{6} + 117328 p T^{7} + 49975 p^{2} T^{8} + 980 p^{3} T^{9} + 330 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 16 T + 394 T^{2} - 3768 T^{3} + 54471 T^{4} - 373912 T^{5} + 4520764 T^{6} - 373912 p T^{7} + 54471 p^{2} T^{8} - 3768 p^{3} T^{9} + 394 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 14 T + 138 T^{2} - 794 T^{3} + 8943 T^{4} - 14476 T^{5} + 53964 T^{6} - 14476 p T^{7} + 8943 p^{2} T^{8} - 794 p^{3} T^{9} + 138 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 - 616 T^{2} + 194482 T^{4} - 40827432 T^{6} + 6291690159 T^{8} - 745855504816 T^{10} + 69562353246940 T^{12} - 745855504816 p^{2} T^{14} + 6291690159 p^{4} T^{16} - 40827432 p^{6} T^{18} + 194482 p^{8} T^{20} - 616 p^{10} T^{22} + p^{12} T^{24} \)
89 \( 1 - 604 T^{2} + 157342 T^{4} - 21524796 T^{6} + 1310341923 T^{8} + 40008531704 T^{10} - 11522524660868 T^{12} + 40008531704 p^{2} T^{14} + 1310341923 p^{4} T^{16} - 21524796 p^{6} T^{18} + 157342 p^{8} T^{20} - 604 p^{10} T^{22} + p^{12} T^{24} \)
97 \( 1 - 664 T^{2} + 229302 T^{4} - 53558632 T^{6} + 9319581603 T^{8} - 1265176645904 T^{10} + 136977060412524 T^{12} - 1265176645904 p^{2} T^{14} + 9319581603 p^{4} T^{16} - 53558632 p^{6} T^{18} + 229302 p^{8} T^{20} - 664 p^{10} T^{22} + p^{12} T^{24} \)
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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

−2.42970159983223813770563573606, −2.26382975131538263798396161922, −2.25036390338312590380558439262, −2.13714517098447414705344395622, −2.11709967254400859871954123100, −1.99659966350430822472974890173, −1.98608776848252407738303607983, −1.77723365497177063603607282720, −1.65556293435415831872503169555, −1.51218324382522046662720764015, −1.41105921582217648055880046634, −1.40509150887942997853386664511, −1.33505322268717074173876646288, −1.32645054045206644075754557124, −1.31272074718391264656227916979, −1.28357170064498520238122638004, −1.12167734581721745009995037813, −0.856873143756245486405282074858, −0.840357051893539556461435379470, −0.75478949611862918045126438782, −0.67147440344093626947873014202, −0.42198887051908162004788314493, −0.36631579523645178015097081061, −0.26808568078045948933149337001, −0.13981930741998909380532053128, 0.13981930741998909380532053128, 0.26808568078045948933149337001, 0.36631579523645178015097081061, 0.42198887051908162004788314493, 0.67147440344093626947873014202, 0.75478949611862918045126438782, 0.840357051893539556461435379470, 0.856873143756245486405282074858, 1.12167734581721745009995037813, 1.28357170064498520238122638004, 1.31272074718391264656227916979, 1.32645054045206644075754557124, 1.33505322268717074173876646288, 1.40509150887942997853386664511, 1.41105921582217648055880046634, 1.51218324382522046662720764015, 1.65556293435415831872503169555, 1.77723365497177063603607282720, 1.98608776848252407738303607983, 1.99659966350430822472974890173, 2.11709967254400859871954123100, 2.13714517098447414705344395622, 2.25036390338312590380558439262, 2.26382975131538263798396161922, 2.42970159983223813770563573606

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

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