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Properties

Label 2-959-959.510-c0-0-0
Degree $2$
Conductor $959$
Sign $0.658 + 0.752i$
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  + (−1.58 − 0.614i)2-s + (1.39 + 1.27i)4-s + (0.602 + 0.798i)7-s + (−0.675 − 1.35i)8-s + (0.850 − 0.526i)9-s + (−1.37 − 1.25i)11-s + (−0.465 − 1.63i)14-s + (0.0633 + 0.683i)16-s + (−1.67 + 0.312i)18-s + (1.41 + 2.83i)22-s + (1.98 − 0.183i)23-s + (−0.739 + 0.673i)25-s + (−0.174 + 1.88i)28-s + (0.719 − 0.0666i)29-s + (−0.0955 + 0.335i)32-s + ⋯
L(s)  = 1  + (−1.58 − 0.614i)2-s + (1.39 + 1.27i)4-s + (0.602 + 0.798i)7-s + (−0.675 − 1.35i)8-s + (0.850 − 0.526i)9-s + (−1.37 − 1.25i)11-s + (−0.465 − 1.63i)14-s + (0.0633 + 0.683i)16-s + (−1.67 + 0.312i)18-s + (1.41 + 2.83i)22-s + (1.98 − 0.183i)23-s + (−0.739 + 0.673i)25-s + (−0.174 + 1.88i)28-s + (0.719 − 0.0666i)29-s + (−0.0955 + 0.335i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(959\)    =    \(7 \cdot 137\)
Sign: $0.658 + 0.752i$
Analytic conductor: \(0.478603\)
Root analytic conductor: \(0.691811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{959} (510, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 959,\ (\ :0),\ 0.658 + 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5182455311\)
\(L(\frac12)\) \(\approx\) \(0.5182455311\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.602 - 0.798i)T \)
137 \( 1 + T \)
good2 \( 1 + (1.58 + 0.614i)T + (0.739 + 0.673i)T^{2} \)
3 \( 1 + (-0.850 + 0.526i)T^{2} \)
5 \( 1 + (0.739 - 0.673i)T^{2} \)
11 \( 1 + (1.37 + 1.25i)T + (0.0922 + 0.995i)T^{2} \)
13 \( 1 + (-0.273 + 0.961i)T^{2} \)
17 \( 1 + (0.602 + 0.798i)T^{2} \)
19 \( 1 + (-0.932 + 0.361i)T^{2} \)
23 \( 1 + (-1.98 + 0.183i)T + (0.982 - 0.183i)T^{2} \)
29 \( 1 + (-0.719 + 0.0666i)T + (0.982 - 0.183i)T^{2} \)
31 \( 1 + (0.932 + 0.361i)T^{2} \)
37 \( 1 - 1.47T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.365 + 1.95i)T + (-0.932 + 0.361i)T^{2} \)
47 \( 1 + (0.445 - 0.895i)T^{2} \)
53 \( 1 + (-0.353 - 1.89i)T + (-0.932 + 0.361i)T^{2} \)
59 \( 1 + (-0.445 + 0.895i)T^{2} \)
61 \( 1 + (-0.445 - 0.895i)T^{2} \)
67 \( 1 + (-0.840 + 0.634i)T + (0.273 - 0.961i)T^{2} \)
71 \( 1 + (0.907 + 0.995i)T + (-0.0922 + 0.995i)T^{2} \)
73 \( 1 + (0.273 + 0.961i)T^{2} \)
79 \( 1 + (0.353 + 0.100i)T + (0.850 + 0.526i)T^{2} \)
83 \( 1 + (-0.602 + 0.798i)T^{2} \)
89 \( 1 + (0.739 - 0.673i)T^{2} \)
97 \( 1 + (0.0922 + 0.995i)T^{2} \)
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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

−10.12219137069263262222299223123, −9.151517491140932864292318530534, −8.702840646327426865981259310367, −7.86042108502743514751612619094, −7.20446493244033060872944712114, −5.93920543307328235472486691681, −4.90416027331911709642361772269, −3.25317472524266985254007404373, −2.39181598818617889686586792535, −1.03341601508565436157433883195, 1.25835708961806479113656621189, 2.43333732129233661685461638866, 4.42507053541835621432463139836, 5.15402998183969486976897968964, 6.61455553699729144529405442737, 7.32522228380410421753503761263, 7.78420212032438486813986258174, 8.457536766415387194107151027667, 9.785659217755793604791269786709, 9.979744260200630306390684877798

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