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Properties

Label 2-490-1.1-c1-0-11
Degree 22
Conductor 490490
Sign 11
Analytic cond. 3.912663.91266
Root an. cond. 1.978041.97804
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 8-s + 9-s + 10-s + 3·11-s + 2·12-s − 5·13-s + 2·15-s + 16-s − 6·17-s + 18-s + 19-s + 20-s + 3·22-s + 3·23-s + 2·24-s + 25-s − 5·26-s − 4·27-s − 6·29-s + 2·30-s + 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.577·12-s − 1.38·13-s + 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.639·22-s + 0.625·23-s + 0.408·24-s + 1/5·25-s − 0.980·26-s − 0.769·27-s − 1.11·29-s + 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯

Functional equation

Λ(s)=(490s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(490s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 3.912663.91266
Root analytic conductor: 1.978041.97804
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 490, ( :1/2), 1)(2,\ 490,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0927935543.092793554
L(12)L(\frac12) ≈\approx 3.0927935543.092793554
L(32)L(\frac{3}{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−T 1 - T
5 1−T 1 - T
7 1 1
good3 1−2T+pT2 1 - 2 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−11T+pT2 1 - 11 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1−3T+pT2 1 - 3 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+14T+pT2 1 + 14 T + p 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

−11.14507794859466678682828912363, −9.808705402590549548523141273172, −9.263851768447220146642287460280, −8.289976858527741605754067242124, −7.22555857938969291949122618245, −6.41172881162855055433374816405, −5.10121929575789116373472283151, −4.09357447562145058841583904603, −2.89619370582642367784100306287, −2.01395492425859231821166088910, 2.01395492425859231821166088910, 2.89619370582642367784100306287, 4.09357447562145058841583904603, 5.10121929575789116373472283151, 6.41172881162855055433374816405, 7.22555857938969291949122618245, 8.289976858527741605754067242124, 9.263851768447220146642287460280, 9.808705402590549548523141273172, 11.14507794859466678682828912363

Graph of the ZZ-function along the critical line