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Properties

Label 2-490-1.1-c5-0-35
Degree 22
Conductor 490490
Sign −1-1
Analytic cond. 78.588078.5880
Root an. cond. 8.864998.86499
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 5·3-s + 16·4-s − 25·5-s + 20·6-s − 64·8-s − 218·9-s + 100·10-s + 198·11-s − 80·12-s − 340·13-s + 125·15-s + 256·16-s + 1.84e3·17-s + 872·18-s − 1.21e3·19-s − 400·20-s − 792·22-s + 2.82e3·23-s + 320·24-s + 625·25-s + 1.36e3·26-s + 2.30e3·27-s − 4.53e3·29-s − 500·30-s − 712·31-s − 1.02e3·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.320·3-s + 1/2·4-s − 0.447·5-s + 0.226·6-s − 0.353·8-s − 0.897·9-s + 0.316·10-s + 0.493·11-s − 0.160·12-s − 0.557·13-s + 0.143·15-s + 1/4·16-s + 1.55·17-s + 0.634·18-s − 0.768·19-s − 0.223·20-s − 0.348·22-s + 1.11·23-s + 0.113·24-s + 1/5·25-s + 0.394·26-s + 0.608·27-s − 1.00·29-s − 0.101·30-s − 0.133·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(490s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(490s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 490490    =    2⋅5⋅722 \cdot 5 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 78.588078.5880
Root analytic conductor: 8.864998.86499
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 490, ( :5/2), −1)(2,\ 490,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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
5 1+p2T 1 + p^{2} T
7 1 1
good3 1+5T+p5T2 1 + 5 T + p^{5} T^{2}
11 1−18pT+p5T2 1 - 18 p T + p^{5} T^{2}
13 1+340T+p5T2 1 + 340 T + p^{5} T^{2}
17 1−1848T+p5T2 1 - 1848 T + p^{5} T^{2}
19 1+1210T+p5T2 1 + 1210 T + p^{5} T^{2}
23 1−2823T+p5T2 1 - 2823 T + p^{5} T^{2}
29 1+4539T+p5T2 1 + 4539 T + p^{5} T^{2}
31 1+712T+p5T2 1 + 712 T + p^{5} T^{2}
37 1+7324T+p5T2 1 + 7324 T + p^{5} T^{2}
41 1−15633T+p5T2 1 - 15633 T + p^{5} T^{2}
43 1−15827T+p5T2 1 - 15827 T + p^{5} T^{2}
47 1+3192T+p5T2 1 + 3192 T + p^{5} T^{2}
53 1+20046T+p5T2 1 + 20046 T + p^{5} T^{2}
59 1−23046T+p5T2 1 - 23046 T + p^{5} T^{2}
61 1+379T+p5T2 1 + 379 T + p^{5} T^{2}
67 1+35473T+p5T2 1 + 35473 T + p^{5} T^{2}
71 1−71814T+p5T2 1 - 71814 T + p^{5} T^{2}
73 1−31664T+p5T2 1 - 31664 T + p^{5} T^{2}
79 1−8534T+p5T2 1 - 8534 T + p^{5} T^{2}
83 1+106551T+p5T2 1 + 106551 T + p^{5} T^{2}
89 1+12303T+p5T2 1 + 12303 T + p^{5} T^{2}
97 1+102802T+p5T2 1 + 102802 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

−9.615870674164418658863117830815, −8.871562357472780033269785115168, −7.930195139847184114257882536551, −7.14990445350704787974502615024, −6.05986591078844456835430972606, −5.15054978845127285450422846705, −3.70984437855153237747120140118, −2.60339062353864051727882878826, −1.10518105160673696012672215949, 0, 1.10518105160673696012672215949, 2.60339062353864051727882878826, 3.70984437855153237747120140118, 5.15054978845127285450422846705, 6.05986591078844456835430972606, 7.14990445350704787974502615024, 7.930195139847184114257882536551, 8.871562357472780033269785115168, 9.615870674164418658863117830815

Graph of the ZZ-function along the critical line