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Properties

Label 2-869-869.157-c0-0-1
Degree $2$
Conductor $869$
Sign $0.655 - 0.755i$
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  + (−0.866 + 0.629i)2-s + (0.0458 − 0.141i)4-s + (−1.41 − 1.03i)5-s + (−0.282 − 0.867i)8-s + (−0.809 + 0.587i)9-s + 1.87·10-s + (0.535 + 0.844i)11-s + (1.03 − 0.749i)13-s + (0.911 + 0.662i)16-s + (0.331 − 1.01i)18-s + (0.450 + 1.38i)19-s + (−0.210 + 0.152i)20-s + (−0.996 − 0.394i)22-s + 1.45·23-s + (0.640 + 1.97i)25-s + (−0.422 + 1.29i)26-s + ⋯
L(s)  = 1  + (−0.866 + 0.629i)2-s + (0.0458 − 0.141i)4-s + (−1.41 − 1.03i)5-s + (−0.282 − 0.867i)8-s + (−0.809 + 0.587i)9-s + 1.87·10-s + (0.535 + 0.844i)11-s + (1.03 − 0.749i)13-s + (0.911 + 0.662i)16-s + (0.331 − 1.01i)18-s + (0.450 + 1.38i)19-s + (−0.210 + 0.152i)20-s + (−0.996 − 0.394i)22-s + 1.45·23-s + (0.640 + 1.97i)25-s + (−0.422 + 1.29i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $0.655 - 0.755i$
Analytic conductor: \(0.433687\)
Root analytic conductor: \(0.658549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{869} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 869,\ (\ :0),\ 0.655 - 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4709761521\)
\(L(\frac12)\) \(\approx\) \(0.4709761521\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.535 - 0.844i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (1.41 + 1.03i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.03 + 0.749i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.45T + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-1.50 + 1.09i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + 0.374T + T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.263 - 0.809i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 - 1.93T + T^{2} \)
97 \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
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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

−10.28525766066797671766918245685, −9.279969913525273664350950539070, −8.539641499652409876233935021862, −8.032686944840645017517574594755, −7.52865329453520918882018071847, −6.37075458247018337521396068690, −5.22546651144970450619600551112, −4.16406764218319890375043261541, −3.30365295213831375178309489017, −1.03889774240273580466912475836, 0.889662271639472934932589881810, 2.91928111320206341655274156436, 3.37181503313948160286045968467, 4.74419491286681973010414489617, 6.21877177800301261599284433453, 6.85223172072686921557647690387, 7.991614813591565222193887346137, 8.829024836155047620224899820932, 9.137214106156333698445876358627, 10.55686292839872533604222653086

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