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Properties

Label 2-869-869.394-c0-0-0
Degree $2$
Conductor $869$
Sign $-0.521 - 0.853i$
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  + (−0.263 + 0.809i)2-s + (0.222 + 0.161i)4-s + (0.331 + 1.01i)5-s + (−0.878 + 0.638i)8-s + (0.309 − 0.951i)9-s − 0.912·10-s + (−0.425 + 0.904i)11-s + (−0.115 + 0.356i)13-s + (−0.200 − 0.617i)16-s + (0.688 + 0.500i)18-s + (−0.101 + 0.0738i)19-s + (−0.0910 + 0.280i)20-s + (−0.620 − 0.582i)22-s + 0.125·23-s + (−0.120 + 0.0872i)25-s + (−0.258 − 0.187i)26-s + ⋯
L(s)  = 1  + (−0.263 + 0.809i)2-s + (0.222 + 0.161i)4-s + (0.331 + 1.01i)5-s + (−0.878 + 0.638i)8-s + (0.309 − 0.951i)9-s − 0.912·10-s + (−0.425 + 0.904i)11-s + (−0.115 + 0.356i)13-s + (−0.200 − 0.617i)16-s + (0.688 + 0.500i)18-s + (−0.101 + 0.0738i)19-s + (−0.0910 + 0.280i)20-s + (−0.620 − 0.582i)22-s + 0.125·23-s + (−0.120 + 0.0872i)25-s + (−0.258 − 0.187i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $-0.521 - 0.853i$
Analytic conductor: \(0.433687\)
Root analytic conductor: \(0.658549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{869} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 869,\ (\ :0),\ -0.521 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9608508398\)
\(L(\frac12)\) \(\approx\) \(0.9608508398\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.425 - 0.904i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.263 - 0.809i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.115 - 0.356i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.101 - 0.0738i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - 0.125T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.450 + 1.38i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + 1.85T + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - 1.75T + T^{2} \)
97 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.49808222952940201131839827685, −9.709017833124843879119395038556, −8.961177836533504389009987410449, −7.80437604820464926458442542550, −7.17266096644870657146129865968, −6.50061514338033425127027462578, −5.85132434324563329891801850550, −4.44760168909230943961778375461, −3.16464825335285970316204618076, −2.19890560075597474813551305929, 1.11424292704568100250996511998, 2.27147826417288976803704682619, 3.37793155177891328252425937204, 4.84325020704586923065868408539, 5.50818997928742753321105372463, 6.55540913070250209180522833895, 7.74473111739701357645026837829, 8.637095575351563692571519408228, 9.285987206819581927536750986292, 10.40099236174471925028640486213

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