Properties

Label 2-490-35.13-c1-0-6
Degree $2$
Conductor $490$
Sign $0.922 - 0.385i$
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 + (−0.204 + 0.204i)3-s − 1.00i·4-s + (−1.42 + 1.71i)5-s + 0.289i·6-s + (−0.707 − 0.707i)8-s + 2.91i·9-s + (0.204 + 2.22i)10-s + 5.62·11-s + (0.204 + 0.204i)12-s + (−1.42 + 1.42i)13-s + (−0.0593 − 0.645i)15-s − 1.00·16-s + (3.75 + 3.75i)17-s + (2.06 + 2.06i)18-s + 3.89·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.118 + 0.118i)3-s − 0.500i·4-s + (−0.639 + 0.768i)5-s + 0.118i·6-s + (−0.250 − 0.250i)8-s + 0.972i·9-s + (0.0647 + 0.704i)10-s + 1.69·11-s + (0.0591 + 0.0591i)12-s + (−0.396 + 0.396i)13-s + (−0.0153 − 0.166i)15-s − 0.250·16-s + (0.910 + 0.910i)17-s + (0.486 + 0.486i)18-s + 0.892·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.922 - 0.385i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.922 - 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62579 + 0.325545i\)
\(L(\frac12)\) \(\approx\) \(1.62579 + 0.325545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 1.71i)T \)
7 \( 1 \)
good3 \( 1 + (0.204 - 0.204i)T - 3iT^{2} \)
11 \( 1 - 5.62T + 11T^{2} \)
13 \( 1 + (1.42 - 1.42i)T - 13iT^{2} \)
17 \( 1 + (-3.75 - 3.75i)T + 17iT^{2} \)
19 \( 1 - 3.89T + 19T^{2} \)
23 \( 1 + (0.794 + 0.794i)T + 23iT^{2} \)
29 \( 1 - 3.15iT - 29T^{2} \)
31 \( 1 - 3.84iT - 31T^{2} \)
37 \( 1 + (3.56 - 3.56i)T - 37iT^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (-1.85 - 1.85i)T + 43iT^{2} \)
47 \( 1 + (4.16 + 4.16i)T + 47iT^{2} \)
53 \( 1 + (0.978 + 0.978i)T + 53iT^{2} \)
59 \( 1 + 5.47T + 59T^{2} \)
61 \( 1 + 4.60iT - 61T^{2} \)
67 \( 1 + (-0.597 + 0.597i)T - 67iT^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + (-3.96 + 3.96i)T - 73iT^{2} \)
79 \( 1 - 6.24iT - 79T^{2} \)
83 \( 1 + (-5.67 + 5.67i)T - 83iT^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + (6.63 + 6.63i)T + 97iT^{2} \)
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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

−11.07732351249585290465039556568, −10.38115296675572096377736561517, −9.480872089176598134697731430135, −8.296902501141151134836452576810, −7.22461563533522279531240877613, −6.39442485721068598750632337097, −5.17803520422210065590873906949, −4.05830956953643844815848353309, −3.24391352131379105315704167814, −1.68933632908054256151371260313, 0.995629204024991212220061288654, 3.31605583529956669112137306465, 4.13756846431422879147871955410, 5.24610974381491038728894302443, 6.23891513932917386384107612771, 7.22492049188136751115102996980, 8.003405031671787871798077951387, 9.225430513000438587950694694867, 9.584400064819730628856028730774, 11.40166952456431935627777065568

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