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Properties

Label 2-959-959.209-c0-0-0
Degree $2$
Conductor $959$
Sign $0.989 + 0.144i$
Analytic cond. $0.478603$
Root an. cond. $0.691811$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0822 + 0.165i)2-s + (0.582 − 0.770i)4-s + (0.932 − 0.361i)7-s + (0.356 + 0.0666i)8-s + (0.0922 + 0.995i)9-s + (−0.537 + 0.711i)11-s + (0.136 + 0.124i)14-s + (−0.246 − 0.864i)16-s + (−0.156 + 0.0971i)18-s + (−0.161 − 0.0302i)22-s + (0.149 + 0.526i)23-s + (−0.602 − 0.798i)25-s + (0.264 − 0.929i)28-s + (−0.243 − 0.857i)29-s + (0.390 − 0.356i)32-s + ⋯
L(s)  = 1  + (0.0822 + 0.165i)2-s + (0.582 − 0.770i)4-s + (0.932 − 0.361i)7-s + (0.356 + 0.0666i)8-s + (0.0922 + 0.995i)9-s + (−0.537 + 0.711i)11-s + (0.136 + 0.124i)14-s + (−0.246 − 0.864i)16-s + (−0.156 + 0.0971i)18-s + (−0.161 − 0.0302i)22-s + (0.149 + 0.526i)23-s + (−0.602 − 0.798i)25-s + (0.264 − 0.929i)28-s + (−0.243 − 0.857i)29-s + (0.390 − 0.356i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(959\)    =    \(7 \cdot 137\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(0.478603\)
Root analytic conductor: \(0.691811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{959} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 959,\ (\ :0),\ 0.989 + 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.282900739\)
\(L(\frac12)\) \(\approx\) \(1.282900739\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.932 + 0.361i)T \)
137 \( 1 - T \)
good2 \( 1 + (-0.0822 - 0.165i)T + (-0.602 + 0.798i)T^{2} \)
3 \( 1 + (-0.0922 - 0.995i)T^{2} \)
5 \( 1 + (0.602 + 0.798i)T^{2} \)
11 \( 1 + (0.537 - 0.711i)T + (-0.273 - 0.961i)T^{2} \)
13 \( 1 + (-0.739 + 0.673i)T^{2} \)
17 \( 1 + (-0.932 + 0.361i)T^{2} \)
19 \( 1 + (-0.445 + 0.895i)T^{2} \)
23 \( 1 + (-0.149 - 0.526i)T + (-0.850 + 0.526i)T^{2} \)
29 \( 1 + (0.243 + 0.857i)T + (-0.850 + 0.526i)T^{2} \)
31 \( 1 + (-0.445 - 0.895i)T^{2} \)
37 \( 1 + 1.20T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.465 + 0.288i)T + (0.445 - 0.895i)T^{2} \)
47 \( 1 + (0.982 - 0.183i)T^{2} \)
53 \( 1 + (1.25 - 0.778i)T + (0.445 - 0.895i)T^{2} \)
59 \( 1 + (0.982 - 0.183i)T^{2} \)
61 \( 1 + (0.982 + 0.183i)T^{2} \)
67 \( 1 + (-0.172 + 0.0666i)T + (0.739 - 0.673i)T^{2} \)
71 \( 1 + (-0.726 - 0.961i)T + (-0.273 + 0.961i)T^{2} \)
73 \( 1 + (-0.739 - 0.673i)T^{2} \)
79 \( 1 + (1.25 - 1.14i)T + (0.0922 - 0.995i)T^{2} \)
83 \( 1 + (-0.932 - 0.361i)T^{2} \)
89 \( 1 + (0.602 + 0.798i)T^{2} \)
97 \( 1 + (0.273 + 0.961i)T^{2} \)
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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

−10.31500421753947949316196159810, −9.639128296775415486604777364679, −8.285925119650276255958621088237, −7.62033310248967554702830124090, −6.96863361869093115400458131166, −5.74957524608088118547386484332, −5.05329065982907345844057633324, −4.26009337931226431059781600916, −2.45758340788483198912308802544, −1.63131935126081512579444194753, 1.68463116918141560616737166162, 2.95641208187315200095737973134, 3.77839171537650792828913882031, 4.97584374981650875220125180346, 5.99345116879002379148107133150, 6.94338947054700591180935217812, 7.79422288114880171478846021902, 8.525063682180666107484020450572, 9.245201668131820078890990144288, 10.50675769651923813705392462036

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