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Properties

Label 2-869-869.631-c0-0-2
Degree $2$
Conductor $869$
Sign $-0.515 + 0.856i$
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  + (−1.56 − 1.13i)2-s + (0.850 + 2.61i)4-s + (1.60 − 1.16i)5-s + (1.04 − 3.22i)8-s + (−0.809 − 0.587i)9-s − 3.84·10-s + (0.968 + 0.248i)11-s + (−0.866 − 0.629i)13-s + (−3.09 + 2.24i)16-s + (0.598 + 1.84i)18-s + (−0.115 + 0.356i)19-s + (4.41 + 3.21i)20-s + (−1.23 − 1.49i)22-s − 0.374·23-s + (0.907 − 2.79i)25-s + (0.641 + 1.97i)26-s + ⋯
L(s)  = 1  + (−1.56 − 1.13i)2-s + (0.850 + 2.61i)4-s + (1.60 − 1.16i)5-s + (1.04 − 3.22i)8-s + (−0.809 − 0.587i)9-s − 3.84·10-s + (0.968 + 0.248i)11-s + (−0.866 − 0.629i)13-s + (−3.09 + 2.24i)16-s + (0.598 + 1.84i)18-s + (−0.115 + 0.356i)19-s + (4.41 + 3.21i)20-s + (−1.23 − 1.49i)22-s − 0.374·23-s + (0.907 − 2.79i)25-s + (0.641 + 1.97i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $-0.515 + 0.856i$
Analytic conductor: \(0.433687\)
Root analytic conductor: \(0.658549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{869} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 869,\ (\ :0),\ -0.515 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5893024592\)
\(L(\frac12)\) \(\approx\) \(0.5893024592\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.968 - 0.248i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.866 + 0.629i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.115 - 0.356i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + 0.374T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + 0.851T + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.541 - 1.66i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 - 0.125T + T^{2} \)
97 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.834075546300960520217777096878, −9.428552750621193949120419321273, −8.724882365508504639522742263586, −8.136917783688276932861761898162, −6.79686181198633638179306338355, −5.87262427641128419735989705087, −4.52508676160837799960325548454, −3.06314371282818272096183079348, −2.05045027916809333370138674364, −1.04080633391758185682198524141, 1.77787488811084348890061319681, 2.61432916058158973326059900528, 5.04713466749610905939924793121, 5.97211997926535628630405314611, 6.47443193987193210834511946828, 7.14058431713484979649916453237, 8.132421937399356696833545731043, 9.152869485997483538988976866054, 9.568846174802213670711065021162, 10.29249392824807761077650306787

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