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Properties

Label 2-490-35.17-c1-0-1
Degree $2$
Conductor $490$
Sign $0.124 - 0.992i$
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 + (0.279 − 0.0749i)3-s + (−0.866 − 0.499i)4-s + (−0.774 + 2.09i)5-s − 0.289i·6-s + (−0.707 + 0.707i)8-s + (−2.52 + 1.45i)9-s + (1.82 + 1.29i)10-s + (−2.81 + 4.87i)11-s + (−0.279 − 0.0749i)12-s + (−1.42 − 1.42i)13-s + (−0.0593 + 0.645i)15-s + (0.500 + 0.866i)16-s + (1.37 + 5.12i)17-s + (0.754 + 2.81i)18-s + (−1.94 − 3.37i)19-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.161 − 0.0432i)3-s + (−0.433 − 0.249i)4-s + (−0.346 + 0.938i)5-s − 0.118i·6-s + (−0.249 + 0.249i)8-s + (−0.841 + 0.486i)9-s + (0.577 + 0.408i)10-s + (−0.848 + 1.46i)11-s + (−0.0807 − 0.0216i)12-s + (−0.396 − 0.396i)13-s + (−0.0153 + 0.166i)15-s + (0.125 + 0.216i)16-s + (0.333 + 1.24i)17-s + (0.177 + 0.663i)18-s + (−0.446 − 0.773i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.124 - 0.992i$
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: \((2,\ 490,\ (\ :1/2),\ 0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.634957 + 0.560079i\)
\(L(\frac12)\) \(\approx\) \(0.634957 + 0.560079i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.774 - 2.09i)T \)
7 \( 1 \)
good3 \( 1 + (-0.279 + 0.0749i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (2.81 - 4.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.42 + 1.42i)T + 13iT^{2} \)
17 \( 1 + (-1.37 - 5.12i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.94 + 3.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.08 - 0.290i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.15iT - 29T^{2} \)
31 \( 1 + (-3.33 - 1.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.30 - 4.86i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.21iT - 41T^{2} \)
43 \( 1 + (-1.85 + 1.85i)T - 43iT^{2} \)
47 \( 1 + (-5.69 - 1.52i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.357 + 1.33i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.73 + 4.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.99 + 2.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.816 - 0.218i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + (5.42 - 1.45i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.41 - 3.12i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.67 - 5.67i)T + 83iT^{2} \)
89 \( 1 + (5.96 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
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

−11.05221557230308722424118757564, −10.40515857653778619153825325455, −9.749166073886091358815466853923, −8.367939668894665519456500768880, −7.69110220236342881161448238454, −6.58560799506489642996672303134, −5.34366359865727667099260966279, −4.31082098408524957146596343395, −2.97215954207746584910328850678, −2.21021609317869410576305427698, 0.45101872867046694541920494957, 2.89215487418815193672837700046, 4.04472808469236889806323780820, 5.31366675997893676429624434325, 5.82451156727485875779369323831, 7.19928141295368388781705707491, 8.181527757643798815702177087552, 8.740628907050548837521234108370, 9.514005788560946301656154769367, 10.84429238333211818887802887013

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