Properties

Label 2-869-869.236-c0-0-2
Degree $2$
Conductor $869$
Sign $0.923 - 0.383i$
Analytic cond. $0.433687$
Root an. cond. $0.658549$
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  + (−0.115 − 0.356i)2-s + (0.695 − 0.505i)4-s + (−0.393 + 1.21i)5-s + (−0.563 − 0.409i)8-s + (0.309 + 0.951i)9-s + 0.477·10-s + (−0.187 + 0.982i)11-s + (0.450 + 1.38i)13-s + (0.184 − 0.569i)16-s + (0.303 − 0.220i)18-s + (−1.56 − 1.13i)19-s + (0.338 + 1.04i)20-s + (0.371 − 0.0469i)22-s + 1.93·23-s + (−0.505 − 0.367i)25-s + (0.442 − 0.321i)26-s + ⋯
L(s)  = 1  + (−0.115 − 0.356i)2-s + (0.695 − 0.505i)4-s + (−0.393 + 1.21i)5-s + (−0.563 − 0.409i)8-s + (0.309 + 0.951i)9-s + 0.477·10-s + (−0.187 + 0.982i)11-s + (0.450 + 1.38i)13-s + (0.184 − 0.569i)16-s + (0.303 − 0.220i)18-s + (−1.56 − 1.13i)19-s + (0.338 + 1.04i)20-s + (0.371 − 0.0469i)22-s + 1.93·23-s + (−0.505 − 0.367i)25-s + (0.442 − 0.321i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $0.923 - 0.383i$
Analytic conductor: \(0.433687\)
Root analytic conductor: \(0.658549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{869} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 869,\ (\ :0),\ 0.923 - 0.383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.014078670\)
\(L(\frac12)\) \(\approx\) \(1.014078670\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.187 - 0.982i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - 1.93T + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.613 + 1.88i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - 0.125T + T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-1.50 + 1.09i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + 0.851T + T^{2} \)
97 \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \)
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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

−10.76456863361160350925629690455, −9.761841725756523863113519743693, −8.936255710551546332362228435487, −7.56274519182694084835807218583, −6.91421293125189772042793580231, −6.47725464079317745532104205442, −5.03099235916851967993349007389, −4.01567302380562497444522141501, −2.60527746224242096665925153334, −1.98258010952793066961948010652, 1.15179391303695001800305845858, 3.04990361758688374312513655076, 3.81763044149622388577638137818, 5.16666649573776746886430860571, 6.02429196586892523821345388082, 6.88384502289417794830297230295, 7.978443370659723474111494026631, 8.556608637949895889920796865397, 9.026456553954469796358144830968, 10.51093014150006132857870614151

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