Properties

Label 2-959-959.622-c0-0-0
Degree $2$
Conductor $959$
Sign $0.848 - 0.529i$
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 + (1.93 − 0.361i)23-s + (0.0922 − 0.995i)25-s + (0.201 + 0.0377i)28-s + (−1.45 + 0.271i)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 + (1.93 − 0.361i)23-s + (0.0922 − 0.995i)25-s + (0.201 + 0.0377i)28-s + (−1.45 + 0.271i)29-s + (−0.343 − 0.212i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(959\)    =    \(7 \cdot 137\)
Sign: $0.848 - 0.529i$
Analytic conductor: \(0.478603\)
Root analytic conductor: \(0.691811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{959} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 959,\ (\ :0),\ 0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.446676858\)
\(L(\frac12)\) \(\approx\) \(1.446676858\)
\(L(1)\) 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 \( 1 + (-1.93 + 0.361i)T + (0.932 - 0.361i)T^{2} \)
29 \( 1 + (1.45 - 0.271i)T + (0.932 - 0.361i)T^{2} \)
31 \( 1 + (-0.739 - 0.673i)T^{2} \)
37 \( 1 - 0.184T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1.83 - 0.710i)T + (0.739 - 0.673i)T^{2} \)
47 \( 1 + (0.602 + 0.798i)T^{2} \)
53 \( 1 + (1.58 - 0.614i)T + (0.739 - 0.673i)T^{2} \)
59 \( 1 + (0.602 + 0.798i)T^{2} \)
61 \( 1 + (0.602 - 0.798i)T^{2} \)
67 \( 1 + (0.243 - 0.857i)T + (-0.850 - 0.526i)T^{2} \)
71 \( 1 + (-0.0170 + 0.183i)T + (-0.982 - 0.183i)T^{2} \)
73 \( 1 + (0.850 - 0.526i)T^{2} \)
79 \( 1 + (1.58 + 0.981i)T + (0.445 + 0.895i)T^{2} \)
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

−10.02880834724837163147481424060, −9.546237700171635133747719934980, −8.772071377830495189926301399570, −7.39070339944372702486159517184, −6.76675186664851831392866074409, −6.07991710518340094432023255187, −5.03627441895267124741674889662, −4.40111146751712444663067890565, −3.14935891975581625073973036980, −1.64455450703384481158843432246, 1.56090659480716741238821935335, 3.10516539960428226410965441935, 3.65003529157974893754020891282, 4.76200557177515501377807311968, 5.51421614724102852269041596975, 6.88642206244739624509902099071, 7.60472020705138615601143125593, 8.416741721548940799283038641250, 9.409696072369702339188749573895, 10.51842707603876208901668330875

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