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Properties

Label 2-959-959.734-c0-0-0
Degree $2$
Conductor $959$
Sign $0.810 + 0.585i$
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  + (−1.25 − 0.778i)2-s + (0.527 + 1.06i)4-s + (0.982 − 0.183i)7-s + (0.0251 − 0.271i)8-s + (−0.739 − 0.673i)9-s + (0.757 + 1.52i)11-s + (−1.37 − 0.533i)14-s + (0.470 − 0.623i)16-s + (0.404 + 1.42i)18-s + (0.231 − 2.50i)22-s + (1.27 + 0.961i)23-s + (−0.445 + 0.895i)25-s + (0.713 + 0.945i)28-s + (−0.840 − 0.634i)29-s + (−1.33 + 0.515i)32-s + ⋯
L(s)  = 1  + (−1.25 − 0.778i)2-s + (0.527 + 1.06i)4-s + (0.982 − 0.183i)7-s + (0.0251 − 0.271i)8-s + (−0.739 − 0.673i)9-s + (0.757 + 1.52i)11-s + (−1.37 − 0.533i)14-s + (0.470 − 0.623i)16-s + (0.404 + 1.42i)18-s + (0.231 − 2.50i)22-s + (1.27 + 0.961i)23-s + (−0.445 + 0.895i)25-s + (0.713 + 0.945i)28-s + (−0.840 − 0.634i)29-s + (−1.33 + 0.515i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(959\)    =    \(7 \cdot 137\)
Sign: $0.810 + 0.585i$
Analytic conductor: \(0.478603\)
Root analytic conductor: \(0.691811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{959} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 959,\ (\ :0),\ 0.810 + 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5912943758\)
\(L(\frac12)\) \(\approx\) \(0.5912943758\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.982 + 0.183i)T \)
137 \( 1 + T \)
good2 \( 1 + (1.25 + 0.778i)T + (0.445 + 0.895i)T^{2} \)
3 \( 1 + (0.739 + 0.673i)T^{2} \)
5 \( 1 + (0.445 - 0.895i)T^{2} \)
11 \( 1 + (-0.757 - 1.52i)T + (-0.602 + 0.798i)T^{2} \)
13 \( 1 + (0.932 - 0.361i)T^{2} \)
17 \( 1 + (0.982 - 0.183i)T^{2} \)
19 \( 1 + (0.850 - 0.526i)T^{2} \)
23 \( 1 + (-1.27 - 0.961i)T + (0.273 + 0.961i)T^{2} \)
29 \( 1 + (0.840 + 0.634i)T + (0.273 + 0.961i)T^{2} \)
31 \( 1 + (-0.850 - 0.526i)T^{2} \)
37 \( 1 - 0.891T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.53 + 0.436i)T + (0.850 - 0.526i)T^{2} \)
47 \( 1 + (0.0922 + 0.995i)T^{2} \)
53 \( 1 + (-0.694 + 0.197i)T + (0.850 - 0.526i)T^{2} \)
59 \( 1 + (-0.0922 - 0.995i)T^{2} \)
61 \( 1 + (-0.0922 + 0.995i)T^{2} \)
67 \( 1 + (0.247 + 1.32i)T + (-0.932 + 0.361i)T^{2} \)
71 \( 1 + (1.60 + 0.798i)T + (0.602 + 0.798i)T^{2} \)
73 \( 1 + (-0.932 - 0.361i)T^{2} \)
79 \( 1 + (0.694 + 1.79i)T + (-0.739 + 0.673i)T^{2} \)
83 \( 1 + (-0.982 - 0.183i)T^{2} \)
89 \( 1 + (0.445 - 0.895i)T^{2} \)
97 \( 1 + (-0.602 + 0.798i)T^{2} \)
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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

−10.02393277773664871275769182755, −9.185705056540732215370607609848, −8.963619909851171609152205240684, −7.64405405139459577738644434629, −7.32559558687381755356200575336, −5.87011453049766257039672607713, −4.77800319885045172429509617562, −3.59215042434579919609159617030, −2.24962939761678538199063142631, −1.28452899262356262498712570862, 1.09050692018812924224439802498, 2.69092146065958371812964747243, 4.18823758684871186475470045706, 5.51528911454882148126684345740, 6.12188352377373997262408254223, 7.19966071002271126763029591311, 8.011582002499144583793501885719, 8.728226896541134786182012363144, 8.937380862036685718863619181012, 10.21603645196031664034611734955

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