Properties

Label 2-869-869.394-c0-0-4
Degree $2$
Conductor $869$
Sign $-0.972 + 0.232i$
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  + (0.450 − 1.38i)2-s + (−0.910 − 0.661i)4-s + (−0.574 − 1.76i)5-s + (−0.148 + 0.107i)8-s + (0.309 − 0.951i)9-s − 2.71·10-s + (0.728 + 0.684i)11-s + (−0.613 + 1.88i)13-s + (−0.265 − 0.816i)16-s + (−1.17 − 0.856i)18-s + (−0.866 + 0.629i)19-s + (−0.646 + 1.99i)20-s + (1.27 − 0.702i)22-s + 1.07·23-s + (−1.98 + 1.44i)25-s + (2.34 + 1.70i)26-s + ⋯
L(s)  = 1  + (0.450 − 1.38i)2-s + (−0.910 − 0.661i)4-s + (−0.574 − 1.76i)5-s + (−0.148 + 0.107i)8-s + (0.309 − 0.951i)9-s − 2.71·10-s + (0.728 + 0.684i)11-s + (−0.613 + 1.88i)13-s + (−0.265 − 0.816i)16-s + (−1.17 − 0.856i)18-s + (−0.866 + 0.629i)19-s + (−0.646 + 1.99i)20-s + (1.27 − 0.702i)22-s + 1.07·23-s + (−1.98 + 1.44i)25-s + (2.34 + 1.70i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $-0.972 + 0.232i$
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: \((2,\ 869,\ (\ :0),\ -0.972 + 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.151952758\)
\(L(\frac12)\) \(\approx\) \(1.151952758\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.728 - 0.684i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-0.450 + 1.38i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - 1.07T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 - 1.93T + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + 0.374T + T^{2} \)
97 \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.728618885739768697760874218946, −9.420653516879937837791747119625, −8.742078511208344147601464570465, −7.46950130392146725999804423987, −6.45555956879143442403078090552, −4.94347646534228555793947934513, −4.21268666013945840663620444448, −3.93172242933181099076793543865, −2.08493534200915078985676191043, −1.12987056450597520703206116259, 2.63346744609971456760910063111, 3.56078025551242301361432562019, 4.79976916359181485400511344396, 5.67481424378641626843375832750, 6.73728004498547108478824921843, 7.06515069330101932701598102572, 7.943836647320558442959048355531, 8.526118408321618985353522655045, 10.15081031400687888130434791883, 10.75822767066634528324959999864

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