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Properties

Label 2-650-65.44-c2-0-0
Degree $2$
Conductor $650$
Sign $0.800 - 0.599i$
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 − 5.69i·3-s + 2i·4-s + (−5.69 + 5.69i)6-s + (0.00593 − 0.00593i)7-s + (2 − 2i)8-s − 23.4·9-s + (−7.25 − 7.25i)11-s + 11.3·12-s + (0.345 + 12.9i)13-s − 0.0118·14-s − 4·16-s − 11.5·17-s + (23.4 + 23.4i)18-s + (−4.06 + 4.06i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s − 1.89i·3-s + 0.5i·4-s + (−0.949 + 0.949i)6-s + (0.000848 − 0.000848i)7-s + (0.250 − 0.250i)8-s − 2.60·9-s + (−0.659 − 0.659i)11-s + 0.949·12-s + (0.0265 + 0.999i)13-s − 0.000848·14-s − 0.250·16-s − 0.681·17-s + (1.30 + 1.30i)18-s + (−0.213 + 0.213i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(17.7112\)
Root analytic conductor: \(4.20846\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1),\ 0.800 - 0.599i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1350140193\)
\(L(\frac12)\) \(\approx\) \(0.1350140193\)
\(L(2)\) 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 \( 1 + (-0.345 - 12.9i)T \)
good3 \( 1 + 5.69iT - 9T^{2} \)
7 \( 1 + (-0.00593 + 0.00593i)T - 49iT^{2} \)
11 \( 1 + (7.25 + 7.25i)T + 121iT^{2} \)
17 \( 1 + 11.5T + 289T^{2} \)
19 \( 1 + (4.06 - 4.06i)T - 361iT^{2} \)
23 \( 1 - 34.5T + 529T^{2} \)
29 \( 1 + 4.46T + 841T^{2} \)
31 \( 1 + (30.7 - 30.7i)T - 961iT^{2} \)
37 \( 1 + (-29.1 + 29.1i)T - 1.36e3iT^{2} \)
41 \( 1 + (30.9 - 30.9i)T - 1.68e3iT^{2} \)
43 \( 1 + 59.6T + 1.84e3T^{2} \)
47 \( 1 + (14.8 - 14.8i)T - 2.20e3iT^{2} \)
53 \( 1 + 59.4iT - 2.80e3T^{2} \)
59 \( 1 + (-55.9 - 55.9i)T + 3.48e3iT^{2} \)
61 \( 1 + 99.2T + 3.72e3T^{2} \)
67 \( 1 + (-33.6 - 33.6i)T + 4.48e3iT^{2} \)
71 \( 1 + (-70.6 + 70.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (-70.2 + 70.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 88.4T + 6.24e3T^{2} \)
83 \( 1 + (-19.5 - 19.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (106. + 106. i)T + 7.92e3iT^{2} \)
97 \( 1 + (13.9 + 13.9i)T + 9.40e3iT^{2} \)
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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

−10.84539668876952742141871535725, −9.264906124685635777643897590500, −8.609787002844717814363759900404, −7.83847731437098792209050255456, −6.97533356283469796757362425793, −6.36117949595797244289422193415, −5.10458903025395794814956304968, −3.28271046047989295336793752679, −2.25840913338195897925737414054, −1.27605676889770940797397275194, 0.05977839141325588454317792149, 2.61213003739110847909117397039, 3.79090756531755150065407054575, 4.98192521126052337951981822095, 5.33650715846907310289997093131, 6.64708891232399235009405655469, 7.918469407995772893823508091446, 8.703180501029039997903432923464, 9.502059169071890441967605815881, 10.10418799649308688823605768667

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