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Properties

Label 24-585e12-1.1-c1e12-0-0
Degree $24$
Conductor $1.606\times 10^{33}$
Sign $1$
Analytic cond. $1.07943\times 10^{8}$
Root an. cond. $2.16130$
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  − 4·4-s + 6·5-s + 6·16-s + 12·19-s − 24·20-s + 17·25-s − 18·29-s − 16·31-s − 14·41-s − 18·49-s + 4·59-s + 6·61-s + 6·64-s + 12·71-s − 48·76-s − 104·79-s + 36·80-s − 20·89-s + 72·95-s − 68·100-s + 26·101-s + 24·109-s + 72·116-s + 40·121-s + 64·124-s + 38·125-s + 127-s + ⋯
L(s)  = 1  − 2·4-s + 2.68·5-s + 3/2·16-s + 2.75·19-s − 5.36·20-s + 17/5·25-s − 3.34·29-s − 2.87·31-s − 2.18·41-s − 2.57·49-s + 0.520·59-s + 0.768·61-s + 3/4·64-s + 1.42·71-s − 5.50·76-s − 11.7·79-s + 4.02·80-s − 2.11·89-s + 7.38·95-s − 6.79·100-s + 2.58·101-s + 2.29·109-s + 6.68·116-s + 3.63·121-s + 5.74·124-s + 3.39·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{24} \cdot 5^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(1.07943\times 10^{8}\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{24} \cdot 5^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9070279446\)
\(L(\frac12)\) \(\approx\) \(0.9070279446\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 - 3 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 15 T^{2} + 3 p T^{4} - 322 T^{6} + 3 p^{3} T^{8} - 15 p^{4} T^{10} + p^{6} T^{12} \)
good2 \( 1 + p^{2} T^{2} + 5 p T^{4} + 5 p T^{6} - p^{4} T^{8} - 43 p T^{10} - 223 T^{12} - 43 p^{3} T^{14} - p^{8} T^{16} + 5 p^{7} T^{18} + 5 p^{9} T^{20} + p^{12} T^{22} + p^{12} T^{24} \)
7 \( 1 + 18 T^{2} + 82 T^{4} + 536 T^{6} + 12326 T^{8} + 78550 T^{10} + 235838 T^{12} + 78550 p^{2} T^{14} + 12326 p^{4} T^{16} + 536 p^{6} T^{18} + 82 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \)
11 \( ( 1 - 20 T^{2} + 16 T^{3} + 180 T^{4} - 160 T^{5} - 1674 T^{6} - 160 p T^{7} + 180 p^{2} T^{8} + 16 p^{3} T^{9} - 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 67 T^{2} + 2371 T^{4} + 56020 T^{6} + 996641 T^{8} + 14527513 T^{10} + 221018582 T^{12} + 14527513 p^{2} T^{14} + 996641 p^{4} T^{16} + 56020 p^{6} T^{18} + 2371 p^{8} T^{20} + 67 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 - 6 T - 20 T^{2} + 100 T^{3} + 764 T^{4} - 1606 T^{5} - 11338 T^{6} - 1606 p T^{7} + 764 p^{2} T^{8} + 100 p^{3} T^{9} - 20 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 126 T^{2} + 9010 T^{4} + 451400 T^{6} + 17445734 T^{8} + 540254266 T^{10} + 13705453502 T^{12} + 540254266 p^{2} T^{14} + 17445734 p^{4} T^{16} + 451400 p^{6} T^{18} + 9010 p^{8} T^{20} + 126 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{6} \)
31 \( ( 1 + 4 T + 53 T^{2} + 288 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
37 \( 1 + 187 T^{2} + 19451 T^{4} + 1426940 T^{6} + 81658601 T^{8} + 3849109833 T^{10} + 153778997622 T^{12} + 3849109833 p^{2} T^{14} + 81658601 p^{4} T^{16} + 1426940 p^{6} T^{18} + 19451 p^{8} T^{20} + 187 p^{10} T^{22} + p^{12} T^{24} \)
41 \( ( 1 + 7 T - 45 T^{2} - 500 T^{3} + 929 T^{4} + 12237 T^{5} + 28438 T^{6} + 12237 p T^{7} + 929 p^{2} T^{8} - 500 p^{3} T^{9} - 45 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
43 \( 1 + 178 T^{2} + 17426 T^{4} + 1084520 T^{6} + 47574566 T^{8} + 1584620742 T^{10} + 56532979182 T^{12} + 1584620742 p^{2} T^{14} + 47574566 p^{4} T^{16} + 1084520 p^{6} T^{18} + 17426 p^{8} T^{20} + 178 p^{10} T^{22} + p^{12} T^{24} \)
47 \( ( 1 - 46 T^{2} + 3407 T^{4} - 54276 T^{6} + 3407 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 147 T^{2} + 6923 T^{4} - 205346 T^{6} + 6923 p^{2} T^{8} - 147 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 - 2 T - 2 p T^{2} - 44 T^{3} + 7486 T^{4} + 9630 T^{5} - 471322 T^{6} + 9630 p T^{7} + 7486 p^{2} T^{8} - 44 p^{3} T^{9} - 2 p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 3 T - 125 T^{2} + 100 T^{3} + 9029 T^{4} + 4247 T^{5} - 611842 T^{6} + 4247 p T^{7} + 9029 p^{2} T^{8} + 100 p^{3} T^{9} - 125 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 302 T^{2} + 48066 T^{4} + 5726680 T^{6} + 564790166 T^{8} + 46598346138 T^{10} + 3315667632142 T^{12} + 46598346138 p^{2} T^{14} + 564790166 p^{4} T^{16} + 5726680 p^{6} T^{18} + 48066 p^{8} T^{20} + 302 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 - 6 T - 176 T^{2} + 380 T^{3} + 24936 T^{4} - 24734 T^{5} - 1974402 T^{6} - 24734 p T^{7} + 24936 p^{2} T^{8} + 380 p^{3} T^{9} - 176 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 223 T^{2} + 31055 T^{4} - 2685330 T^{6} + 31055 p^{2} T^{8} - 223 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 + 26 T + 417 T^{2} + 4268 T^{3} + 417 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( ( 1 - 222 T^{2} + 35303 T^{4} - 3305156 T^{6} + 35303 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 + 10 T - 10 T^{2} + 8 p T^{3} + 370 T^{4} - 95250 T^{5} - 421426 T^{6} - 95250 p T^{7} + 370 p^{2} T^{8} + 8 p^{4} T^{9} - 10 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( 1 + 302 T^{2} + 44346 T^{4} + 4094200 T^{6} + 2434958 p T^{8} - 3936373062 T^{10} - 1781970067778 T^{12} - 3936373062 p^{2} T^{14} + 2434958 p^{5} T^{16} + 4094200 p^{6} T^{18} + 44346 p^{8} T^{20} + 302 p^{10} T^{22} + p^{12} T^{24} \)
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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

−3.57727177497259332258716230240, −3.54124645348894457645338672372, −3.43708033704031881580614468253, −3.10549081509625716194294002069, −2.96133333651627281707083712305, −2.96046448366631143807703748647, −2.87450166157146223744966424214, −2.80348721626429249239269928139, −2.77250270747782686014676919063, −2.64703013165463467774925814111, −2.57876203079263162072378718853, −2.30172091280879352053246176522, −1.94489639391473782580204083805, −1.92774809208598142110591799140, −1.91344589134087193778022050199, −1.71674320660499278305593747918, −1.69862365798905887183173390712, −1.68584599691798898852558225611, −1.49442307084098923167413859699, −1.31273820435065317068897950309, −1.17564206107977052159223287520, −1.04071622953006736883957638714, −0.55234955533891122290873390927, −0.37296955454060633829143896227, −0.14730469439585271047318327932, 0.14730469439585271047318327932, 0.37296955454060633829143896227, 0.55234955533891122290873390927, 1.04071622953006736883957638714, 1.17564206107977052159223287520, 1.31273820435065317068897950309, 1.49442307084098923167413859699, 1.68584599691798898852558225611, 1.69862365798905887183173390712, 1.71674320660499278305593747918, 1.91344589134087193778022050199, 1.92774809208598142110591799140, 1.94489639391473782580204083805, 2.30172091280879352053246176522, 2.57876203079263162072378718853, 2.64703013165463467774925814111, 2.77250270747782686014676919063, 2.80348721626429249239269928139, 2.87450166157146223744966424214, 2.96046448366631143807703748647, 2.96133333651627281707083712305, 3.10549081509625716194294002069, 3.43708033704031881580614468253, 3.54124645348894457645338672372, 3.57727177497259332258716230240

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

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