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Properties

Label 2-882-1.1-c5-0-20
Degree 22
Conductor 882882
Sign 11
Analytic cond. 141.458141.458
Root an. cond. 11.893611.8936
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  − 4·2-s + 16·4-s − 54·5-s − 64·8-s + 216·10-s + 594·11-s − 26·13-s + 256·16-s + 534·17-s + 3.00e3·19-s − 864·20-s − 2.37e3·22-s + 3.51e3·23-s − 209·25-s + 104·26-s + 4.29e3·29-s − 8.03e3·31-s − 1.02e3·32-s − 2.13e3·34-s − 502·37-s − 1.20e4·38-s + 3.45e3·40-s − 9.87e3·41-s + 9.06e3·43-s + 9.50e3·44-s − 1.40e4·46-s − 1.14e3·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.965·5-s − 0.353·8-s + 0.683·10-s + 1.48·11-s − 0.0426·13-s + 1/4·16-s + 0.448·17-s + 1.90·19-s − 0.482·20-s − 1.04·22-s + 1.38·23-s − 0.0668·25-s + 0.0301·26-s + 0.948·29-s − 1.50·31-s − 0.176·32-s − 0.316·34-s − 0.0602·37-s − 1.34·38-s + 0.341·40-s − 0.916·41-s + 0.747·43-s + 0.740·44-s − 0.978·46-s − 0.0752·47-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 141.458141.458
Root analytic conductor: 11.893611.8936
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 882, ( :5/2), 1)(2,\ 882,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.5611790781.561179078
L(12)L(\frac12) ≈\approx 1.5611790781.561179078
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+p2T 1 + p^{2} T
3 1 1
7 1 1
good5 1+54T+p5T2 1 + 54 T + p^{5} T^{2}
11 1−54pT+p5T2 1 - 54 p T + p^{5} T^{2}
13 1+2pT+p5T2 1 + 2 p T + p^{5} T^{2}
17 1−534T+p5T2 1 - 534 T + p^{5} T^{2}
19 1−3004T+p5T2 1 - 3004 T + p^{5} T^{2}
23 1−3510T+p5T2 1 - 3510 T + p^{5} T^{2}
29 1−4296T+p5T2 1 - 4296 T + p^{5} T^{2}
31 1+8036T+p5T2 1 + 8036 T + p^{5} T^{2}
37 1+502T+p5T2 1 + 502 T + p^{5} T^{2}
41 1+9870T+p5T2 1 + 9870 T + p^{5} T^{2}
43 1−9068T+p5T2 1 - 9068 T + p^{5} T^{2}
47 1+1140T+p5T2 1 + 1140 T + p^{5} T^{2}
53 1−28356T+p5T2 1 - 28356 T + p^{5} T^{2}
59 1−8196T+p5T2 1 - 8196 T + p^{5} T^{2}
61 1+29822T+p5T2 1 + 29822 T + p^{5} T^{2}
67 1+62884T+p5T2 1 + 62884 T + p^{5} T^{2}
71 1+34398T+p5T2 1 + 34398 T + p^{5} T^{2}
73 1+56990T+p5T2 1 + 56990 T + p^{5} T^{2}
79 1−49496T+p5T2 1 - 49496 T + p^{5} T^{2}
83 1−52512T+p5T2 1 - 52512 T + p^{5} T^{2}
89 1−48282T+p5T2 1 - 48282 T + p^{5} T^{2}
97 1−83938T+p5T2 1 - 83938 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

−9.189476434028089374223568211108, −8.757167882597757485016931686790, −7.48424986587415938352901067551, −7.27058394465797468843504994687, −6.11614400893696706878541128583, −4.98669429633174983089053266085, −3.77811131965039058122906361263, −3.08445562069931535246207199051, −1.46505499338608114097251893203, −0.68648713253101715572545763388, 0.68648713253101715572545763388, 1.46505499338608114097251893203, 3.08445562069931535246207199051, 3.77811131965039058122906361263, 4.98669429633174983089053266085, 6.11614400893696706878541128583, 7.27058394465797468843504994687, 7.48424986587415938352901067551, 8.757167882597757485016931686790, 9.189476434028089374223568211108

Graph of the ZZ-function along the critical line