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Properties

Label 2-490-5.2-c2-0-31
Degree $2$
Conductor $490$
Sign $-0.767 + 0.640i$
Analytic cond. $13.3515$
Root an. cond. $3.65397$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−3.73 + 3.73i)3-s + 2i·4-s + (3 − 4i)5-s + 7.47·6-s + (2 − 2i)8-s − 18.9i·9-s + (−7 + i)10-s + 3.47·11-s + (−7.47 − 7.47i)12-s + (−10.9 + 10.9i)13-s + (3.73 + 26.1i)15-s − 4·16-s + (6.95 + 6.95i)17-s + (−18.9 + 18.9i)18-s + 18.4i·19-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−1.24 + 1.24i)3-s + 0.5i·4-s + (0.600 − 0.800i)5-s + 1.24·6-s + (0.250 − 0.250i)8-s − 2.10i·9-s + (−0.700 + 0.100i)10-s + 0.316·11-s + (−0.623 − 0.623i)12-s + (−0.842 + 0.842i)13-s + (0.249 + 1.74i)15-s − 0.250·16-s + (0.409 + 0.409i)17-s + (−1.05 + 1.05i)18-s + 0.970i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.767 + 0.640i$
Analytic conductor: \(13.3515\)
Root analytic conductor: \(3.65397\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1),\ -0.767 + 0.640i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1712231897\)
\(L(\frac12)\) \(\approx\) \(0.1712231897\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 + (-3 + 4i)T \)
7 \( 1 \)
good3 \( 1 + (3.73 - 3.73i)T - 9iT^{2} \)
11 \( 1 - 3.47T + 121T^{2} \)
13 \( 1 + (10.9 - 10.9i)T - 169iT^{2} \)
17 \( 1 + (-6.95 - 6.95i)T + 289iT^{2} \)
19 \( 1 - 18.4iT - 361T^{2} \)
23 \( 1 + (12.6 - 12.6i)T - 529iT^{2} \)
29 \( 1 + 49.8iT - 841T^{2} \)
31 \( 1 + 16T + 961T^{2} \)
37 \( 1 + (25.0 + 25.0i)T + 1.36e3iT^{2} \)
41 \( 1 - 6.04T + 1.68e3T^{2} \)
43 \( 1 + (9.64 - 9.64i)T - 1.84e3iT^{2} \)
47 \( 1 + (14.8 + 14.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (8.04 - 8.04i)T - 2.80e3iT^{2} \)
59 \( 1 + 75.4iT - 3.48e3T^{2} \)
61 \( 1 + 102.T + 3.72e3T^{2} \)
67 \( 1 + (22.7 + 22.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 40.3T + 5.04e3T^{2} \)
73 \( 1 + (-62.7 + 62.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 101. iT - 6.24e3T^{2} \)
83 \( 1 + (51.1 - 51.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 82.7iT - 7.92e3T^{2} \)
97 \( 1 + (45.7 + 45.7i)T + 9.40e3iT^{2} \)
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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

−10.23592316675081181816036625345, −9.686241079437798066626982724062, −9.169402302029237100627710168291, −7.896167868346834304438700710581, −6.37500043865595902803334369624, −5.57202092966158737238999378700, −4.58063735207043144001683556442, −3.78160876599562539243999209117, −1.75954229800915925366326563156, −0.10029048112416828042656591394, 1.35195332892823977321699934573, 2.70086385774676791611979642291, 5.07241651970182195789650281594, 5.71346043167155381061514741347, 6.78080273201788085263547500799, 7.06960964583235375055663609220, 8.043808604735607361952876748434, 9.380924478198620335290432780380, 10.43936571747847349494356145905, 10.94369584187482920210086806826

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