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Properties

Label 2-650-65.2-c1-0-8
Degree $2$
Conductor $650$
Sign $0.00940 - 0.999i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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  + (0.5 + 0.866i)2-s + (1.98 + 0.531i)3-s + (−0.499 + 0.866i)4-s + (0.531 + 1.98i)6-s + (2.67 + 1.54i)7-s − 0.999·8-s + (1.05 + 0.609i)9-s + (−0.858 + 3.20i)11-s + (−1.45 + 1.45i)12-s + (−1.43 − 3.30i)13-s + 3.09i·14-s + (−0.5 − 0.866i)16-s + (1.44 + 5.40i)17-s + 1.21i·18-s + (0.296 − 0.0794i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (1.14 + 0.306i)3-s + (−0.249 + 0.433i)4-s + (0.217 + 0.809i)6-s + (1.01 + 0.584i)7-s − 0.353·8-s + (0.351 + 0.203i)9-s + (−0.258 + 0.965i)11-s + (−0.419 + 0.419i)12-s + (−0.397 − 0.917i)13-s + 0.826i·14-s + (−0.125 − 0.216i)16-s + (0.351 + 1.31i)17-s + 0.287i·18-s + (0.0680 − 0.0182i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00940 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00940 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.00940 - 0.999i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.00940 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86359 + 1.84614i\)
\(L(\frac12)\) \(\approx\) \(1.86359 + 1.84614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (1.43 + 3.30i)T \)
good3 \( 1 + (-1.98 - 0.531i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.67 - 1.54i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.858 - 3.20i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.44 - 5.40i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.296 + 0.0794i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.35 + 5.05i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6.38 + 3.68i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.42 - 1.42i)T - 31iT^{2} \)
37 \( 1 + (3.19 - 1.84i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.05 - 2.42i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (9.26 - 2.48i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 5.28iT - 47T^{2} \)
53 \( 1 + (-2.01 + 2.01i)T - 53iT^{2} \)
59 \( 1 + (0.224 + 0.837i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.36 + 4.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.62 + 2.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.79 + 6.68i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 - 4.04iT - 79T^{2} \)
83 \( 1 + 5.45iT - 83T^{2} \)
89 \( 1 + (-7.03 - 1.88i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.07 + 12.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.51597252886191540978257088736, −9.796365189256632630731523939119, −8.623870253531542829537106237408, −8.251242790229907383868376182160, −7.53601899801849326695923745279, −6.26190842023048159718498064738, −5.14356388931791761584564980983, −4.36813181251116186088638028676, −3.13191715693818202241111398177, −2.09613014773916742796736927559, 1.30040169316445446672658488530, 2.53770619919311891945121610259, 3.44379358313239969175541892278, 4.59702000700249379146255453955, 5.52702060573578745499643153113, 7.08309095629580402890644412438, 7.76015632624260345140725978060, 8.718011345520474719819010659241, 9.335297228745494053481769967773, 10.42197103525457351139410786702

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