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Properties

Label 2-490-1.1-c3-0-21
Degree 22
Conductor 490490
Sign −1-1
Analytic cond. 28.910928.9109
Root an. cond. 5.376885.37688
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 4·4-s − 5·5-s + 8·6-s − 8·8-s − 11·9-s + 10·10-s + 60·11-s − 16·12-s − 38·13-s + 20·15-s + 16·16-s − 42·17-s + 22·18-s + 52·19-s − 20·20-s − 120·22-s + 120·23-s + 32·24-s + 25·25-s + 76·26-s + 152·27-s − 234·29-s − 40·30-s + 304·31-s − 32·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.769·3-s + 1/2·4-s − 0.447·5-s + 0.544·6-s − 0.353·8-s − 0.407·9-s + 0.316·10-s + 1.64·11-s − 0.384·12-s − 0.810·13-s + 0.344·15-s + 1/4·16-s − 0.599·17-s + 0.288·18-s + 0.627·19-s − 0.223·20-s − 1.16·22-s + 1.08·23-s + 0.272·24-s + 1/5·25-s + 0.573·26-s + 1.08·27-s − 1.49·29-s − 0.243·30-s + 1.76·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(490s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(490s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 28.910928.9109
Root analytic conductor: 5.376885.37688
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 490, ( :3/2), −1)(2,\ 490,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT 1 + p T
5 1+pT 1 + p T
7 1 1
good3 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
11 1−60T+p3T2 1 - 60 T + p^{3} T^{2}
13 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
17 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
19 1−52T+p3T2 1 - 52 T + p^{3} T^{2}
23 1−120T+p3T2 1 - 120 T + p^{3} T^{2}
29 1+234T+p3T2 1 + 234 T + p^{3} T^{2}
31 1−304T+p3T2 1 - 304 T + p^{3} T^{2}
37 1+106T+p3T2 1 + 106 T + p^{3} T^{2}
41 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
43 1+196T+p3T2 1 + 196 T + p^{3} T^{2}
47 1+336T+p3T2 1 + 336 T + p^{3} T^{2}
53 1−438T+p3T2 1 - 438 T + p^{3} T^{2}
59 1−444T+p3T2 1 - 444 T + p^{3} T^{2}
61 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
67 1+988T+p3T2 1 + 988 T + p^{3} T^{2}
71 1+720T+p3T2 1 + 720 T + p^{3} T^{2}
73 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
79 1+808T+p3T2 1 + 808 T + p^{3} T^{2}
83 1+612T+p3T2 1 + 612 T + p^{3} T^{2}
89 1+1146T+p3T2 1 + 1146 T + p^{3} T^{2}
97 1−70T+p3T2 1 - 70 T + p^{3} 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

−10.08166994191718082599212834472, −9.185738291158917633618622049403, −8.466737067958872578142520875842, −7.20212487840482341692703094069, −6.61046836033739617668425968860, −5.51176278702901317192354142325, −4.34036861271575492935150930333, −2.95885373404145523362266096864, −1.27809098940068457354395916273, 0, 1.27809098940068457354395916273, 2.95885373404145523362266096864, 4.34036861271575492935150930333, 5.51176278702901317192354142325, 6.61046836033739617668425968860, 7.20212487840482341692703094069, 8.466737067958872578142520875842, 9.185738291158917633618622049403, 10.08166994191718082599212834472

Graph of the ZZ-function along the critical line