Properties

Label 24-91e12-1.1-c1e12-0-1
Degree $24$
Conductor $3.225\times 10^{23}$
Sign $1$
Analytic cond. $0.0216681$
Root an. cond. $0.852431$
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  − 4·2-s + 8·4-s − 8·7-s − 12·8-s + 16·9-s − 8·11-s + 32·14-s + 20·16-s − 64·18-s + 32·22-s − 64·28-s − 4·29-s − 40·32-s + 128·36-s + 12·37-s − 64·44-s + 32·49-s − 12·53-s + 96·56-s + 16·58-s − 128·63-s + 72·64-s + 60·67-s − 192·72-s − 48·74-s + 64·77-s − 4·79-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s − 3.02·7-s − 4.24·8-s + 16/3·9-s − 2.41·11-s + 8.55·14-s + 5·16-s − 15.0·18-s + 6.82·22-s − 12.0·28-s − 0.742·29-s − 7.07·32-s + 64/3·36-s + 1.97·37-s − 9.64·44-s + 32/7·49-s − 1.64·53-s + 12.8·56-s + 2.10·58-s − 16.1·63-s + 9·64-s + 7.33·67-s − 22.6·72-s − 5.57·74-s + 7.29·77-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(7^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(0.0216681\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{91} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1275665564\)
\(L(\frac12)\) \(\approx\) \(0.1275665564\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + 8 T + 32 T^{2} + 72 T^{3} - 13 T^{4} - 704 T^{5} - 2624 T^{6} - 704 p T^{7} - 13 p^{2} T^{8} + 72 p^{3} T^{9} + 32 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 20 T^{2} - 165 T^{4} + 40 p^{2} T^{6} - 165 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} \)
good2 \( ( 1 + p T + p T^{2} + p T^{3} + p T^{6} + p^{4} T^{9} + p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \)
3 \( ( 1 - 8 T^{2} + 5 p^{2} T^{4} - 154 T^{6} + 5 p^{4} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
5 \( 1 + T^{4} + 94 T^{8} - 18431 T^{12} + 94 p^{4} T^{16} + p^{8} T^{20} + p^{12} T^{24} \)
11 \( ( 1 + 4 T + 8 T^{2} + 46 T^{3} + 3 p^{2} T^{4} + 82 p T^{5} + 1762 T^{6} + 82 p^{2} T^{7} + 3 p^{4} T^{8} + 46 p^{3} T^{9} + 8 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
17 \( ( 1 + 20 T^{2} + 445 T^{4} + 4010 T^{6} + 445 p^{2} T^{8} + 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( 1 - 355 T^{4} - 95870 T^{8} + 114782513 T^{12} - 95870 p^{4} T^{16} - 355 p^{8} T^{20} + p^{12} T^{24} \)
23 \( ( 1 - 75 T^{2} + 3030 T^{4} - 83635 T^{6} + 3030 p^{2} T^{8} - 75 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 + T + 54 T^{2} - 27 T^{3} + 54 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{4} \)
31 \( 1 - 1423 T^{4} + 233254 T^{8} + 417094841 T^{12} + 233254 p^{4} T^{16} - 1423 p^{8} T^{20} + p^{12} T^{24} \)
37 \( ( 1 - 6 T + 18 T^{2} - 224 T^{3} + 2987 T^{4} - 9502 T^{5} + 28334 T^{6} - 9502 p T^{7} + 2987 p^{2} T^{8} - 224 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 + 6966 T^{4} + 24349871 T^{8} + 51186333364 T^{12} + 24349871 p^{4} T^{16} + 6966 p^{8} T^{20} + p^{12} T^{24} \)
43 \( ( 1 - 63 T^{2} + 4310 T^{4} - 213179 T^{6} + 4310 p^{2} T^{8} - 63 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( 1 - 4499 T^{4} + 232910 p T^{8} - 19979553119 T^{12} + 232910 p^{5} T^{16} - 4499 p^{8} T^{20} + p^{12} T^{24} \)
53 \( ( 1 + 3 T + 104 T^{2} + 431 T^{3} + 104 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
59 \( 1 + 8826 T^{4} + 20574431 T^{8} + 13508299324 T^{12} + 20574431 p^{4} T^{16} + 8826 p^{8} T^{20} + p^{12} T^{24} \)
61 \( ( 1 - 170 T^{2} + 20487 T^{4} - 1428236 T^{6} + 20487 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 30 T + 450 T^{2} - 5098 T^{3} + 47495 T^{4} - 5660 p T^{5} + 668 p^{2} T^{6} - 5660 p^{2} T^{7} + 47495 p^{2} T^{8} - 5098 p^{3} T^{9} + 450 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 + 134 T^{3} + 3423 T^{4} - 21842 T^{5} + 8978 T^{6} - 21842 p T^{7} + 3423 p^{2} T^{8} + 134 p^{3} T^{9} + p^{6} T^{12} )^{2} \)
73 \( 1 - 8035 T^{4} - 27974390 T^{8} + 458366283353 T^{12} - 27974390 p^{4} T^{16} - 8035 p^{8} T^{20} + p^{12} T^{24} \)
79 \( ( 1 + T - 24 T^{2} - 1007 T^{3} - 24 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( 1 - 2655 T^{4} + 72855350 T^{8} - 404915929367 T^{12} + 72855350 p^{4} T^{16} - 2655 p^{8} T^{20} + p^{12} T^{24} \)
89 \( 1 - 3395 T^{4} + 117237082 T^{8} - 235034612951 T^{12} + 117237082 p^{4} T^{16} - 3395 p^{8} T^{20} + p^{12} T^{24} \)
97 \( 1 + 19557 T^{4} + 258666194 T^{8} + 3056968310041 T^{12} + 258666194 p^{4} T^{16} + 19557 p^{8} T^{20} + p^{12} T^{24} \)
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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

−5.37282860250868997087030688576, −5.26248747738750301828740107458, −4.92826436182125146014309806476, −4.67499310526260694379185632790, −4.64867329246519734774806254725, −4.63693595906412472972589541507, −4.47734364083323878121851964951, −4.27092141553524445919487173591, −4.21483089158316522589752718223, −4.04595004440018051472213196042, −3.74843316107361357390718761227, −3.61502473856282790590608499348, −3.51416850862159419630578358451, −3.39253255635364349815800858537, −3.28769654531385555764249880728, −3.25314211905350525024208082722, −2.81146420929664847500361139621, −2.67919913603457126081096348929, −2.34621893281690777347418491708, −2.18440271401638817364380283416, −2.14438079310811671800561183369, −1.95223962279921209108555583373, −1.37484451771622661573916242012, −1.21962236231779655836807211588, −0.816151103817081694719927913812, 0.816151103817081694719927913812, 1.21962236231779655836807211588, 1.37484451771622661573916242012, 1.95223962279921209108555583373, 2.14438079310811671800561183369, 2.18440271401638817364380283416, 2.34621893281690777347418491708, 2.67919913603457126081096348929, 2.81146420929664847500361139621, 3.25314211905350525024208082722, 3.28769654531385555764249880728, 3.39253255635364349815800858537, 3.51416850862159419630578358451, 3.61502473856282790590608499348, 3.74843316107361357390718761227, 4.04595004440018051472213196042, 4.21483089158316522589752718223, 4.27092141553524445919487173591, 4.47734364083323878121851964951, 4.63693595906412472972589541507, 4.64867329246519734774806254725, 4.67499310526260694379185632790, 4.92826436182125146014309806476, 5.26248747738750301828740107458, 5.37282860250868997087030688576

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

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