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Properties

Label 24-15e24-1.1-c1e12-0-0
Degree $24$
Conductor $1.683\times 10^{28}$
Sign $1$
Analytic cond. $1131.13$
Root an. cond. $1.34038$
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  − 4-s + 7·9-s + 4·11-s + 3·16-s − 16·19-s − 14·29-s − 16·31-s − 7·36-s + 26·41-s − 4·44-s − 23·49-s − 4·59-s − 2·61-s + 14·64-s − 40·71-s + 16·76-s + 4·79-s + 15·81-s + 36·89-s + 28·99-s − 12·101-s − 12·109-s + 14·116-s + 54·121-s + 16·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 7/3·9-s + 1.20·11-s + 3/4·16-s − 3.67·19-s − 2.59·29-s − 2.87·31-s − 7/6·36-s + 4.06·41-s − 0.603·44-s − 3.28·49-s − 0.520·59-s − 0.256·61-s + 7/4·64-s − 4.74·71-s + 1.83·76-s + 0.450·79-s + 5/3·81-s + 3.81·89-s + 2.81·99-s − 1.19·101-s − 1.14·109-s + 1.29·116-s + 4.90·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{24} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(1131.13\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{24} \cdot 5^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1520289584\)
\(L(\frac12)\) \(\approx\) \(0.1520289584\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 7 T^{2} + 34 T^{4} - 41 p T^{6} + 34 p^{2} T^{8} - 7 p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 \)
good2 \( 1 + T^{2} - p T^{4} - 19 T^{6} - 7 p T^{8} + 25 T^{10} + 197 T^{12} + 25 p^{2} T^{14} - 7 p^{5} T^{16} - 19 p^{6} T^{18} - p^{9} T^{20} + p^{10} T^{22} + p^{12} T^{24} \)
7 \( 1 + 23 T^{2} + 41 p T^{4} + 1944 T^{6} + 5053 T^{8} - 51983 T^{10} - 624746 T^{12} - 51983 p^{2} T^{14} + 5053 p^{4} T^{16} + 1944 p^{6} T^{18} + 41 p^{9} T^{20} + 23 p^{10} T^{22} + p^{12} T^{24} \)
11 \( ( 1 - 2 T - 21 T^{2} + 14 T^{3} + 26 p T^{4} + 58 T^{5} - 3673 T^{6} + 58 p T^{7} + 26 p^{3} T^{8} + 14 p^{3} T^{9} - 21 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
13 \( 1 + 54 T^{2} + 1581 T^{4} + 30418 T^{6} + 431466 T^{8} + 4991934 T^{10} + 59312613 T^{12} + 4991934 p^{2} T^{14} + 431466 p^{4} T^{16} + 30418 p^{6} T^{18} + 1581 p^{8} T^{20} + 54 p^{10} T^{22} + p^{12} T^{24} \)
17 \( ( 1 - 82 T^{2} + 3087 T^{4} - 67244 T^{6} + 3087 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 + 4 T + 53 T^{2} + 148 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
23 \( 1 + 63 T^{2} + 1143 T^{4} + 30064 T^{6} + 1830141 T^{8} + 38353041 T^{10} + 434719014 T^{12} + 38353041 p^{2} T^{14} + 1830141 p^{4} T^{16} + 30064 p^{6} T^{18} + 1143 p^{8} T^{20} + 63 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 + 7 T - 9 T^{2} - 304 T^{3} - 803 T^{4} + 101 p T^{5} + 36038 T^{6} + 101 p^{2} T^{7} - 803 p^{2} T^{8} - 304 p^{3} T^{9} - 9 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + 8 T + p T^{2} + 208 T^{3} - 158 T^{4} - 7756 T^{5} - 34897 T^{6} - 7756 p T^{7} - 158 p^{2} T^{8} + 208 p^{3} T^{9} + p^{5} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 162 T^{2} + 11847 T^{4} - 534412 T^{6} + 11847 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( ( 1 - 13 T + 27 T^{2} + 292 T^{3} + 445 T^{4} - 22279 T^{5} + 169790 T^{6} - 22279 p T^{7} + 445 p^{2} T^{8} + 292 p^{3} T^{9} + 27 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
43 \( 1 + 150 T^{2} + 13245 T^{4} + 587794 T^{6} + 8508330 T^{8} - 916818210 T^{10} - 62473985643 T^{12} - 916818210 p^{2} T^{14} + 8508330 p^{4} T^{16} + 587794 p^{6} T^{18} + 13245 p^{8} T^{20} + 150 p^{10} T^{22} + p^{12} T^{24} \)
47 \( 1 + 91 T^{2} + 1339 T^{4} - 12268 T^{6} + 3554941 T^{8} - 111607535 T^{10} - 21624150754 T^{12} - 111607535 p^{2} T^{14} + 3554941 p^{4} T^{16} - 12268 p^{6} T^{18} + 1339 p^{8} T^{20} + 91 p^{10} T^{22} + p^{12} T^{24} \)
53 \( ( 1 - 274 T^{2} + 33303 T^{4} - 2287964 T^{6} + 33303 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 + 2 T - 153 T^{2} - 110 T^{3} + 14962 T^{4} + 3194 T^{5} - 1012513 T^{6} + 3194 p T^{7} + 14962 p^{2} T^{8} - 110 p^{3} T^{9} - 153 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + T - 145 T^{2} - 240 T^{3} + 12217 T^{4} + 14087 T^{5} - 812786 T^{6} + 14087 p T^{7} + 12217 p^{2} T^{8} - 240 p^{3} T^{9} - 145 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 203 T^{2} + 14531 T^{4} + 1222044 T^{6} + 166308757 T^{8} + 11407412521 T^{10} + 552166785598 T^{12} + 11407412521 p^{2} T^{14} + 166308757 p^{4} T^{16} + 1222044 p^{6} T^{18} + 14531 p^{8} T^{20} + 203 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 + 10 T + 121 T^{2} + 712 T^{3} + 121 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
73 \( ( 1 - 246 T^{2} + 30015 T^{4} - 2521972 T^{6} + 30015 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 2 T - 149 T^{2} + 374 T^{3} + 10642 T^{4} - 16154 T^{5} - 772597 T^{6} - 16154 p T^{7} + 10642 p^{2} T^{8} + 374 p^{3} T^{9} - 149 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 + 327 T^{2} + 57207 T^{4} + 6488176 T^{6} + 532197333 T^{8} + 34415866233 T^{10} + 2415401371734 T^{12} + 34415866233 p^{2} T^{14} + 532197333 p^{4} T^{16} + 6488176 p^{6} T^{18} + 57207 p^{8} T^{20} + 327 p^{10} T^{22} + p^{12} T^{24} \)
89 \( ( 1 - 3 T + p T^{2} )^{12} \)
97 \( 1 + 186 T^{2} - 1131 T^{4} - 466418 T^{6} + 278880666 T^{8} + 18455439810 T^{10} - 762781062579 T^{12} + 18455439810 p^{2} T^{14} + 278880666 p^{4} T^{16} - 466418 p^{6} T^{18} - 1131 p^{8} T^{20} + 186 p^{10} T^{22} + p^{12} T^{24} \)
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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

−4.21092329718107548473621695374, −4.19526265774768943996549270892, −4.11759900805592214076556964362, −4.05811075202487900426733149399, −3.84658638109977286883962175326, −3.67037633533474885704514514490, −3.64853079067432313172858473617, −3.61518848473022671311554222943, −3.29854387106738523103019707870, −3.12378051812596816588524476749, −3.10011795885534956160573264503, −3.08241207758453772675010584640, −3.03522782412027395945825485058, −2.45914750752995271755829095510, −2.41994403340668963101620121779, −2.19133844309398681077792098830, −2.13285895480032997415493496905, −1.91912018750320778659264059452, −1.90213756689177971949256658658, −1.87336085586554893542473721574, −1.62811490229421640849011235136, −1.18363938648143425188446841993, −1.11819197348752860652791839234, −1.01904915821286136086626410457, −0.089625988368504343775529566565, 0.089625988368504343775529566565, 1.01904915821286136086626410457, 1.11819197348752860652791839234, 1.18363938648143425188446841993, 1.62811490229421640849011235136, 1.87336085586554893542473721574, 1.90213756689177971949256658658, 1.91912018750320778659264059452, 2.13285895480032997415493496905, 2.19133844309398681077792098830, 2.41994403340668963101620121779, 2.45914750752995271755829095510, 3.03522782412027395945825485058, 3.08241207758453772675010584640, 3.10011795885534956160573264503, 3.12378051812596816588524476749, 3.29854387106738523103019707870, 3.61518848473022671311554222943, 3.64853079067432313172858473617, 3.67037633533474885704514514490, 3.84658638109977286883962175326, 4.05811075202487900426733149399, 4.11759900805592214076556964362, 4.19526265774768943996549270892, 4.21092329718107548473621695374

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

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