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Properties

Label 2-392-1.1-c5-0-1
Degree 22
Conductor 392392
Sign 11
Analytic cond. 62.870462.8704
Root an. cond. 7.929087.92908
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 30·3-s − 32·5-s + 657·9-s − 624·11-s + 708·13-s + 960·15-s − 934·17-s − 1.85e3·19-s − 1.12e3·23-s − 2.10e3·25-s − 1.24e4·27-s − 1.17e3·29-s − 2.90e3·31-s + 1.87e4·33-s − 1.24e4·37-s − 2.12e4·39-s − 2.66e3·41-s − 7.14e3·43-s − 2.10e4·45-s + 7.46e3·47-s + 2.80e4·51-s − 2.72e4·53-s + 1.99e4·55-s + 5.57e4·57-s − 2.49e3·59-s + 1.10e4·61-s − 2.26e4·65-s + ⋯
L(s)  = 1  − 1.92·3-s − 0.572·5-s + 2.70·9-s − 1.55·11-s + 1.16·13-s + 1.10·15-s − 0.783·17-s − 1.18·19-s − 0.441·23-s − 0.672·25-s − 3.27·27-s − 0.259·29-s − 0.543·31-s + 2.99·33-s − 1.49·37-s − 2.23·39-s − 0.247·41-s − 0.589·43-s − 1.54·45-s + 0.493·47-s + 1.50·51-s − 1.33·53-s + 0.890·55-s + 2.27·57-s − 0.0931·59-s + 0.381·61-s − 0.665·65-s + ⋯

Functional equation

Λ(s)=(392s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(392s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 392392    =    23⋅722^{3} \cdot 7^{2}
Sign: 11
Analytic conductor: 62.870462.8704
Root analytic conductor: 7.929087.92908
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 392, ( :5/2), 1)(2,\ 392,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.18272335410.1827233541
L(12)L(\frac12) ≈\approx 0.18272335410.1827233541
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1 1
good3 1+10pT+p5T2 1 + 10 p T + p^{5} T^{2}
5 1+32T+p5T2 1 + 32 T + p^{5} T^{2}
11 1+624T+p5T2 1 + 624 T + p^{5} T^{2}
13 1−708T+p5T2 1 - 708 T + p^{5} T^{2}
17 1+934T+p5T2 1 + 934 T + p^{5} T^{2}
19 1+1858T+p5T2 1 + 1858 T + p^{5} T^{2}
23 1+1120T+p5T2 1 + 1120 T + p^{5} T^{2}
29 1+1174T+p5T2 1 + 1174 T + p^{5} T^{2}
31 1+2908T+p5T2 1 + 2908 T + p^{5} T^{2}
37 1+12462T+p5T2 1 + 12462 T + p^{5} T^{2}
41 1+2662T+p5T2 1 + 2662 T + p^{5} T^{2}
43 1+7144T+p5T2 1 + 7144 T + p^{5} T^{2}
47 1−7468T+p5T2 1 - 7468 T + p^{5} T^{2}
53 1+27274T+p5T2 1 + 27274 T + p^{5} T^{2}
59 1+2490T+p5T2 1 + 2490 T + p^{5} T^{2}
61 1−11096T+p5T2 1 - 11096 T + p^{5} T^{2}
67 1−39756T+p5T2 1 - 39756 T + p^{5} T^{2}
71 1+69888T+p5T2 1 + 69888 T + p^{5} T^{2}
73 1+16450T+p5T2 1 + 16450 T + p^{5} T^{2}
79 1−78376T+p5T2 1 - 78376 T + p^{5} T^{2}
83 1+109818T+p5T2 1 + 109818 T + p^{5} T^{2}
89 1−56966T+p5T2 1 - 56966 T + p^{5} T^{2}
97 1−115946T+p5T2 1 - 115946 T + p^{5} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.80813680664656444905063034820, −10.04449151596136854410088847291, −8.519466127068020938121109254686, −7.48141516646993936153544621032, −6.49557287376772363880965973327, −5.70223509003890007831264735791, −4.80240215977684089515205691944, −3.83229749542496796609269453922, −1.80865514151602647500738204428, −0.24488004366244277408035544745, 0.24488004366244277408035544745, 1.80865514151602647500738204428, 3.83229749542496796609269453922, 4.80240215977684089515205691944, 5.70223509003890007831264735791, 6.49557287376772363880965973327, 7.48141516646993936153544621032, 8.519466127068020938121109254686, 10.04449151596136854410088847291, 10.80813680664656444905063034820

Graph of the ZZ-function along the critical line