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Properties

Label 2-490-35.17-c1-0-14
Degree $2$
Conductor $490$
Sign $0.931 - 0.362i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (2.80 − 0.752i)3-s + (−0.866 − 0.499i)4-s + (2.21 − 0.318i)5-s + 2.90i·6-s + (0.707 − 0.707i)8-s + (4.71 − 2.72i)9-s + (−0.264 + 2.22i)10-s + (−1.83 + 3.17i)11-s + (−2.80 − 0.752i)12-s + (−0.830 − 0.830i)13-s + (5.97 − 2.55i)15-s + (0.500 + 0.866i)16-s + (0.204 + 0.761i)17-s + (1.41 + 5.26i)18-s + (−1.09 − 1.89i)19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (1.62 − 0.434i)3-s + (−0.433 − 0.249i)4-s + (0.989 − 0.142i)5-s + 1.18i·6-s + (0.249 − 0.249i)8-s + (1.57 − 0.908i)9-s + (−0.0837 + 0.702i)10-s + (−0.553 + 0.958i)11-s + (−0.810 − 0.217i)12-s + (−0.230 − 0.230i)13-s + (1.54 − 0.660i)15-s + (0.125 + 0.216i)16-s + (0.0494 + 0.184i)17-s + (0.332 + 1.24i)18-s + (−0.251 − 0.434i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.931 - 0.362i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.931 - 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29712 + 0.431586i\)
\(L(\frac12)\) \(\approx\) \(2.29712 + 0.431586i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (-2.21 + 0.318i)T \)
7 \( 1 \)
good3 \( 1 + (-2.80 + 0.752i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.83 - 3.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.830 + 0.830i)T + 13iT^{2} \)
17 \( 1 + (-0.204 - 0.761i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.09 + 1.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.54 + 1.21i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.62iT - 29T^{2} \)
31 \( 1 + (0.0359 + 0.0207i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0664 + 0.248i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.98iT - 41T^{2} \)
43 \( 1 + (0.474 - 0.474i)T - 43iT^{2} \)
47 \( 1 + (6.18 + 1.65i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.04 - 7.64i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.35 - 9.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.72 - 0.996i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.39 + 1.71i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + (9.52 - 2.55i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (11.6 - 6.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.73 - 9.73i)T + 83iT^{2} \)
89 \( 1 + (0.715 + 1.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.16 + 3.16i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.49170972504882528994888120583, −9.806125204197407219301492822576, −9.070097792995411840990881763417, −8.331970545374858681632834202551, −7.47410186485151472017349265914, −6.71981854977468349966883521227, −5.47617713975061554866927312479, −4.25991646024662848582643297092, −2.73707968916509237183615783966, −1.77370554487681631560856920121, 1.85036758420545955029326088327, 2.78358007150621664299395011469, 3.64848498449353600531546137086, 4.92599031970351442128597281117, 6.25655052573943852470717788009, 7.78298542806299752399260517587, 8.400099774491636846609972435347, 9.304751272031305142252851425849, 9.880228140732317229179389203824, 10.54702088043924955498985584434

Graph of the $Z$-function along the critical line