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Properties

Label 2-869-869.157-c0-0-0
Degree $2$
Conductor $869$
Sign $-0.974 + 0.225i$
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  + (−0.101 + 0.0738i)2-s + (−0.304 + 0.936i)4-s + (−1.17 − 0.856i)5-s + (−0.0770 − 0.236i)8-s + (−0.809 + 0.587i)9-s + 0.183·10-s + (0.0627 − 0.998i)11-s + (−1.56 + 1.13i)13-s + (−0.770 − 0.560i)16-s + (0.0388 − 0.119i)18-s + (−0.263 − 0.809i)19-s + (1.16 − 0.843i)20-s + (0.0672 + 0.106i)22-s − 0.851·23-s + (0.347 + 1.07i)25-s + (0.0751 − 0.231i)26-s + ⋯
L(s)  = 1  + (−0.101 + 0.0738i)2-s + (−0.304 + 0.936i)4-s + (−1.17 − 0.856i)5-s + (−0.0770 − 0.236i)8-s + (−0.809 + 0.587i)9-s + 0.183·10-s + (0.0627 − 0.998i)11-s + (−1.56 + 1.13i)13-s + (−0.770 − 0.560i)16-s + (0.0388 − 0.119i)18-s + (−0.263 − 0.809i)19-s + (1.16 − 0.843i)20-s + (0.0672 + 0.106i)22-s − 0.851·23-s + (0.347 + 1.07i)25-s + (0.0751 − 0.231i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $-0.974 + 0.225i$
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: \((2,\ 869,\ (\ :0),\ -0.974 + 0.225i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02019575775\)
\(L(\frac12)\) \(\approx\) \(0.02019575775\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.0627 + 0.998i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (0.101 - 0.0738i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (1.56 - 1.13i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + 0.851T + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 - 1.75T + T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + 1.85T + T^{2} \)
97 \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \)
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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

−11.22842785881524090977529091733, −9.713069388058414040931826059290, −8.786911964232352343180128709478, −8.395703504779656208593070622542, −7.61826736608433896429875486521, −6.82243017116274628690919247198, −5.27732655840321870504319575720, −4.52565313178946332475405234880, −3.67199640473213467400445666893, −2.48902057699292373084761570769, 0.01947391187140906497996998874, 2.26107167974162340489339000407, 3.44179866804417855485800597473, 4.51076678121168308917136405174, 5.52051417029343944299374833109, 6.45796480819857770334734572381, 7.47859210319309920258170642454, 8.042169533271514768765275515184, 9.268363524002847785920423120842, 10.02194297079476302688339457122

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