Properties

Label 2-869-869.394-c0-0-1
Degree $2$
Conductor $869$
Sign $0.650 - 0.759i$
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  + (−0.613 + 1.88i)2-s + (−2.37 − 1.72i)4-s + (0.0388 + 0.119i)5-s + (3.10 − 2.25i)8-s + (0.309 − 0.951i)9-s − 0.249·10-s + (−0.992 − 0.125i)11-s + (0.541 − 1.66i)13-s + (1.44 + 4.46i)16-s + (1.60 + 1.16i)18-s + (1.03 − 0.749i)19-s + (0.113 − 0.350i)20-s + (0.844 − 1.79i)22-s − 1.27·23-s + (0.796 − 0.578i)25-s + (2.81 + 2.04i)26-s + ⋯
L(s)  = 1  + (−0.613 + 1.88i)2-s + (−2.37 − 1.72i)4-s + (0.0388 + 0.119i)5-s + (3.10 − 2.25i)8-s + (0.309 − 0.951i)9-s − 0.249·10-s + (−0.992 − 0.125i)11-s + (0.541 − 1.66i)13-s + (1.44 + 4.46i)16-s + (1.60 + 1.16i)18-s + (1.03 − 0.749i)19-s + (0.113 − 0.350i)20-s + (0.844 − 1.79i)22-s − 1.27·23-s + (0.796 − 0.578i)25-s + (2.81 + 2.04i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $0.650 - 0.759i$
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: \((2,\ 869,\ (\ :0),\ 0.650 - 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5872454501\)
\(L(\frac12)\) \(\approx\) \(0.5872454501\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.992 + 0.125i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.0388 - 0.119i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-1.03 + 0.749i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.27T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.263 - 0.809i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 - 1.07T + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - 1.45T + T^{2} \)
97 \( 1 + (0.574 - 1.76i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.20145981170883591074356740984, −9.387544958759466472878120335315, −8.535677120203300277624523530007, −7.85868658450927597273696124395, −7.16783160333823865056592452183, −6.20469313559688313278882268948, −5.57628297792713428864445363773, −4.70964588012870301912984812928, −3.36160395853242410723242515792, −0.802447250753115764386888569728, 1.59067977021499034058902461662, 2.37418355369437312437999438159, 3.66027011001660338472885311304, 4.49158839911856923318436880281, 5.42364628343432899906019979188, 7.31817189956906469822512012153, 8.055975251092736689435331370270, 8.826278037614936079746675361817, 9.737775839724417128268652892397, 10.21908995136653133963878804532

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