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Properties

Label 2-490-35.27-c1-0-4
Degree $2$
Conductor $490$
Sign $0.101 + 0.994i$
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  + (−0.707 − 0.707i)2-s + (−2.05 − 2.05i)3-s + 1.00i·4-s + (−0.830 + 2.07i)5-s + 2.90i·6-s + (0.707 − 0.707i)8-s + 5.44i·9-s + (2.05 − 0.880i)10-s + 3.67·11-s + (2.05 − 2.05i)12-s + (−0.830 − 0.830i)13-s + (5.97 − 2.55i)15-s − 1.00·16-s + (0.557 − 0.557i)17-s + (3.85 − 3.85i)18-s + 2.18·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−1.18 − 1.18i)3-s + 0.500i·4-s + (−0.371 + 0.928i)5-s + 1.18i·6-s + (0.250 − 0.250i)8-s + 1.81i·9-s + (0.649 − 0.278i)10-s + 1.10·11-s + (0.593 − 0.593i)12-s + (−0.230 − 0.230i)13-s + (1.54 − 0.660i)15-s − 0.250·16-s + (0.135 − 0.135i)17-s + (0.908 − 0.908i)18-s + 0.502·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.101 + 0.994i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.101 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.495228 - 0.447425i\)
\(L(\frac12)\) \(\approx\) \(0.495228 - 0.447425i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.830 - 2.07i)T \)
7 \( 1 \)
good3 \( 1 + (2.05 + 2.05i)T + 3iT^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
13 \( 1 + (0.830 + 0.830i)T + 13iT^{2} \)
17 \( 1 + (-0.557 + 0.557i)T - 17iT^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + (-3.32 + 3.32i)T - 23iT^{2} \)
29 \( 1 - 2.62iT - 29T^{2} \)
31 \( 1 - 0.0415iT - 31T^{2} \)
37 \( 1 + (-0.181 - 0.181i)T + 37iT^{2} \)
41 \( 1 + 8.98iT - 41T^{2} \)
43 \( 1 + (0.474 - 0.474i)T - 43iT^{2} \)
47 \( 1 + (-4.52 + 4.52i)T - 47iT^{2} \)
53 \( 1 + (-5.59 + 5.59i)T - 53iT^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 1.99iT - 61T^{2} \)
67 \( 1 + (4.68 + 4.68i)T + 67iT^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + (-6.97 - 6.97i)T + 73iT^{2} \)
79 \( 1 + 13.4iT - 79T^{2} \)
83 \( 1 + (-9.73 - 9.73i)T + 83iT^{2} \)
89 \( 1 - 1.43T + 89T^{2} \)
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

−10.92155586572711450929899909656, −10.23531607332734209932288211073, −8.969262241156863294201691604211, −7.78085893005407075808190768380, −6.99826726202871046800807434575, −6.49435954308630325905410281034, −5.25670580572747057248241355265, −3.68109943729207470982808172803, −2.22930726183610971488578736264, −0.76536856014748442773618711789, 1.01085069019790722678782605985, 3.81157471993127266119902632140, 4.67346330700888036890705084180, 5.48337105036690996408206992329, 6.36500413068998422923133343503, 7.52181376813003297938140265209, 8.788949297593700374329489017790, 9.437908999375126731355568687422, 10.07011491579553073317988715283, 11.27579181749119356842873416657

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