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Properties

Label 2-650-65.34-c2-0-34
Degree $2$
Conductor $650$
Sign $-0.902 - 0.429i$
Analytic cond. $17.7112$
Root an. cond. $4.20846$
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 − 5.08i·3-s − 2i·4-s + (5.08 + 5.08i)6-s + (4.62 + 4.62i)7-s + (2 + 2i)8-s − 16.8·9-s + (4.44 − 4.44i)11-s − 10.1·12-s + (2.21 + 12.8i)13-s − 9.24·14-s − 4·16-s − 15.2·17-s + (16.8 − 16.8i)18-s + (−24.6 − 24.6i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 1.69i·3-s − 0.5i·4-s + (0.847 + 0.847i)6-s + (0.660 + 0.660i)7-s + (0.250 + 0.250i)8-s − 1.87·9-s + (0.403 − 0.403i)11-s − 0.847·12-s + (0.170 + 0.985i)13-s − 0.660·14-s − 0.250·16-s − 0.896·17-s + (0.936 − 0.936i)18-s + (−1.29 − 1.29i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.902 - 0.429i$
Analytic conductor: \(17.7112\)
Root analytic conductor: \(4.20846\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1),\ -0.902 - 0.429i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2689618740\)
\(L(\frac12)\) \(\approx\) \(0.2689618740\)
\(L(2)\) 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 \( 1 \)
13 \( 1 + (-2.21 - 12.8i)T \)
good3 \( 1 + 5.08iT - 9T^{2} \)
7 \( 1 + (-4.62 - 4.62i)T + 49iT^{2} \)
11 \( 1 + (-4.44 + 4.44i)T - 121iT^{2} \)
17 \( 1 + 15.2T + 289T^{2} \)
19 \( 1 + (24.6 + 24.6i)T + 361iT^{2} \)
23 \( 1 + 25.9T + 529T^{2} \)
29 \( 1 + 18.2T + 841T^{2} \)
31 \( 1 + (28.4 + 28.4i)T + 961iT^{2} \)
37 \( 1 + (-16.1 - 16.1i)T + 1.36e3iT^{2} \)
41 \( 1 + (-30.6 - 30.6i)T + 1.68e3iT^{2} \)
43 \( 1 + 44.2T + 1.84e3T^{2} \)
47 \( 1 + (46.2 + 46.2i)T + 2.20e3iT^{2} \)
53 \( 1 - 75.9iT - 2.80e3T^{2} \)
59 \( 1 + (31.8 - 31.8i)T - 3.48e3iT^{2} \)
61 \( 1 + 4.84T + 3.72e3T^{2} \)
67 \( 1 + (-7.47 + 7.47i)T - 4.48e3iT^{2} \)
71 \( 1 + (33.9 + 33.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (-98.2 - 98.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 38.4T + 6.24e3T^{2} \)
83 \( 1 + (77.7 - 77.7i)T - 6.88e3iT^{2} \)
89 \( 1 + (-20.1 + 20.1i)T - 7.92e3iT^{2} \)
97 \( 1 + (-30.0 + 30.0i)T - 9.40e3iT^{2} \)
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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

−9.442869270784679199205004796109, −8.635248651767530944158771172452, −8.169202470997112650214631276799, −7.10665836754878054343860989365, −6.48698842950911511149881184908, −5.78759235744207139311256882212, −4.40694502860235784829847384274, −2.34743595421105182314269740165, −1.67082198758002177485069756058, −0.10904496788174186542822757143, 1.92987095157916269317032994272, 3.55212023542660872231084181855, 4.12316864093403505994227279029, 5.03193453138854519116911100426, 6.24564641023968359515326479132, 7.74545963010817515624766490440, 8.463294761420190646286864492503, 9.336230693446326300632823693084, 10.12415820356698399982316570815, 10.71449348797602680664200868377

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