Properties

Label 2-959-959.748-c0-0-0
Degree $2$
Conductor $959$
Sign $0.983 + 0.181i$
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  + (0.658 + 0.600i)2-s + (−0.0189 − 0.204i)4-s + (0.273 − 0.961i)7-s + (0.647 − 0.857i)8-s + (−0.445 + 0.895i)9-s + (−0.136 − 1.47i)11-s + (0.757 − 0.469i)14-s + (0.739 − 0.138i)16-s + (−0.831 + 0.322i)18-s + (0.794 − 1.05i)22-s + (0.0675 + 0.361i)23-s + (−0.0922 + 0.995i)25-s + (−0.201 − 0.0377i)28-s + (0.247 + 1.32i)29-s + (−0.343 − 0.212i)32-s + ⋯
L(s)  = 1  + (0.658 + 0.600i)2-s + (−0.0189 − 0.204i)4-s + (0.273 − 0.961i)7-s + (0.647 − 0.857i)8-s + (−0.445 + 0.895i)9-s + (−0.136 − 1.47i)11-s + (0.757 − 0.469i)14-s + (0.739 − 0.138i)16-s + (−0.831 + 0.322i)18-s + (0.794 − 1.05i)22-s + (0.0675 + 0.361i)23-s + (−0.0922 + 0.995i)25-s + (−0.201 − 0.0377i)28-s + (0.247 + 1.32i)29-s + (−0.343 − 0.212i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(959\)    =    \(7 \cdot 137\)
Sign: $0.983 + 0.181i$
Analytic conductor: \(0.478603\)
Root analytic conductor: \(0.691811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{959} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 959,\ (\ :0),\ 0.983 + 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407981284\)
\(L(\frac12)\) \(\approx\) \(1.407981284\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.273 + 0.961i)T \)
137 \( 1 + T \)
good2 \( 1 + (-0.658 - 0.600i)T + (0.0922 + 0.995i)T^{2} \)
3 \( 1 + (0.445 - 0.895i)T^{2} \)
5 \( 1 + (0.0922 - 0.995i)T^{2} \)
11 \( 1 + (0.136 + 1.47i)T + (-0.982 + 0.183i)T^{2} \)
13 \( 1 + (-0.850 - 0.526i)T^{2} \)
17 \( 1 + (0.273 - 0.961i)T^{2} \)
19 \( 1 + (-0.739 + 0.673i)T^{2} \)
23 \( 1 + (-0.0675 - 0.361i)T + (-0.932 + 0.361i)T^{2} \)
29 \( 1 + (-0.247 - 1.32i)T + (-0.932 + 0.361i)T^{2} \)
31 \( 1 + (0.739 + 0.673i)T^{2} \)
37 \( 1 - 0.184T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.132 - 0.342i)T + (-0.739 + 0.673i)T^{2} \)
47 \( 1 + (-0.602 - 0.798i)T^{2} \)
53 \( 1 + (-0.380 - 0.981i)T + (-0.739 + 0.673i)T^{2} \)
59 \( 1 + (0.602 + 0.798i)T^{2} \)
61 \( 1 + (0.602 - 0.798i)T^{2} \)
67 \( 1 + (-1.72 - 0.489i)T + (0.850 + 0.526i)T^{2} \)
71 \( 1 + (1.98 + 0.183i)T + (0.982 + 0.183i)T^{2} \)
73 \( 1 + (0.850 - 0.526i)T^{2} \)
79 \( 1 + (0.380 - 0.614i)T + (-0.445 - 0.895i)T^{2} \)
83 \( 1 + (-0.273 - 0.961i)T^{2} \)
89 \( 1 + (0.0922 - 0.995i)T^{2} \)
97 \( 1 + (-0.982 + 0.183i)T^{2} \)
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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

−10.53357527127165995217390406319, −9.342372421411527344854975672072, −8.322127880325172089722770989964, −7.54429382799342975934011112541, −6.75311216145146555810584167819, −5.69693259551834684783126393301, −5.17541118482741746101220377141, −4.13229213431788235095653279404, −3.11913183834086991641241350184, −1.29554734125878117701305106424, 2.05488769182451142071173634315, 2.79316767997292805511927804283, 4.02641826642534486257199201786, 4.77873454800551382524972450009, 5.75968138826759934566512541877, 6.76608437867381722432931769343, 7.87660688393214193450615078888, 8.580092695307411352860337572945, 9.498800290759996152481825028202, 10.30569977547421852584311520509

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