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Properties

Label 2-490-1.1-c5-0-5
Degree 22
Conductor 490490
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 9·3-s + 16·4-s − 25·5-s − 36·6-s − 64·8-s − 162·9-s + 100·10-s − 187·11-s + 144·12-s − 627·13-s − 225·15-s + 256·16-s − 1.81e3·17-s + 648·18-s − 258·19-s − 400·20-s + 748·22-s + 2.97e3·23-s − 576·24-s + 625·25-s + 2.50e3·26-s − 3.64e3·27-s + 1.29e3·29-s + 900·30-s − 1.91e3·31-s − 1.02e3·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.465·11-s + 0.288·12-s − 1.02·13-s − 0.258·15-s + 1/4·16-s − 1.52·17-s + 0.471·18-s − 0.163·19-s − 0.223·20-s + 0.329·22-s + 1.17·23-s − 0.204·24-s + 1/5·25-s + 0.727·26-s − 0.962·27-s + 0.286·29-s + 0.182·30-s − 0.358·31-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: 11
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: 00
Selberg data: (2, 490, ( :5/2), 1)(2,\ 490,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.92332793830.9233279383
L(12)L(\frac12) ≈\approx 0.92332793830.9233279383
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−p2T+p5T2 1 - p^{2} T + p^{5} T^{2}
11 1+17pT+p5T2 1 + 17 p T + p^{5} T^{2}
13 1+627T+p5T2 1 + 627 T + p^{5} T^{2}
17 1+1813T+p5T2 1 + 1813 T + p^{5} T^{2}
19 1+258T+p5T2 1 + 258 T + p^{5} T^{2}
23 1−2970T+p5T2 1 - 2970 T + p^{5} T^{2}
29 1−1299T+p5T2 1 - 1299 T + p^{5} T^{2}
31 1+1916T+p5T2 1 + 1916 T + p^{5} T^{2}
37 1−6578T+p5T2 1 - 6578 T + p^{5} T^{2}
41 1+6676T+p5T2 1 + 6676 T + p^{5} T^{2}
43 1−3178T+p5T2 1 - 3178 T + p^{5} T^{2}
47 1−22001T+p5T2 1 - 22001 T + p^{5} T^{2}
53 1−26168T+p5T2 1 - 26168 T + p^{5} T^{2}
59 1+3932T+p5T2 1 + 3932 T + p^{5} T^{2}
61 1−48740T+p5T2 1 - 48740 T + p^{5} T^{2}
67 1+44832T+p5T2 1 + 44832 T + p^{5} T^{2}
71 1−63736T+p5T2 1 - 63736 T + p^{5} T^{2}
73 1+60470T+p5T2 1 + 60470 T + p^{5} T^{2}
79 1+43721T+p5T2 1 + 43721 T + p^{5} T^{2}
83 1+1172pT+p5T2 1 + 1172 p T + p^{5} T^{2}
89 1+45560T+p5T2 1 + 45560 T + p^{5} T^{2}
97 1−57295T+p5T2 1 - 57295 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

−10.06946045990873309741306223506, −9.008714711326159717011128606890, −8.576298291749840449319879406820, −7.55116738608735753244862317396, −6.85931802459737413477618739223, −5.52708750154168906910146212207, −4.33216611808213363172542923865, −2.92884824972011609889419047376, −2.21876074194027474424678774896, −0.49528892512479640959912457415, 0.49528892512479640959912457415, 2.21876074194027474424678774896, 2.92884824972011609889419047376, 4.33216611808213363172542923865, 5.52708750154168906910146212207, 6.85931802459737413477618739223, 7.55116738608735753244862317396, 8.576298291749840449319879406820, 9.008714711326159717011128606890, 10.06946045990873309741306223506

Graph of the ZZ-function along the critical line