Properties

Label 2-177-1.1-c13-0-35
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 135.·2-s − 729·3-s + 1.00e4·4-s − 3.67e4·5-s + 9.84e4·6-s − 1.77e5·7-s − 2.52e5·8-s + 5.31e5·9-s + 4.96e6·10-s − 8.38e6·11-s − 7.33e6·12-s + 4.85e6·13-s + 2.39e7·14-s + 2.67e7·15-s − 4.82e7·16-s + 1.51e7·17-s − 7.18e7·18-s − 1.04e8·19-s − 3.69e8·20-s + 1.29e8·21-s + 1.13e9·22-s + 5.03e8·23-s + 1.84e8·24-s + 1.29e8·25-s − 6.55e8·26-s − 3.87e8·27-s − 1.78e9·28-s + ⋯
L(s)  = 1  − 1.49·2-s − 0.577·3-s + 1.22·4-s − 1.05·5-s + 0.861·6-s − 0.569·7-s − 0.341·8-s + 0.333·9-s + 1.57·10-s − 1.42·11-s − 0.709·12-s + 0.278·13-s + 0.850·14-s + 0.607·15-s − 0.719·16-s + 0.152·17-s − 0.497·18-s − 0.510·19-s − 1.29·20-s + 0.328·21-s + 2.12·22-s + 0.708·23-s + 0.196·24-s + 0.106·25-s − 0.416·26-s − 0.192·27-s − 0.699·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 729T \)
59 \( 1 + 4.21e10T \)
good2 \( 1 + 135.T + 8.19e3T^{2} \)
5 \( 1 + 3.67e4T + 1.22e9T^{2} \)
7 \( 1 + 1.77e5T + 9.68e10T^{2} \)
11 \( 1 + 8.38e6T + 3.45e13T^{2} \)
13 \( 1 - 4.85e6T + 3.02e14T^{2} \)
17 \( 1 - 1.51e7T + 9.90e15T^{2} \)
19 \( 1 + 1.04e8T + 4.20e16T^{2} \)
23 \( 1 - 5.03e8T + 5.04e17T^{2} \)
29 \( 1 + 4.08e9T + 1.02e19T^{2} \)
31 \( 1 + 6.12e9T + 2.44e19T^{2} \)
37 \( 1 - 9.81e9T + 2.43e20T^{2} \)
41 \( 1 - 3.36e10T + 9.25e20T^{2} \)
43 \( 1 + 3.54e10T + 1.71e21T^{2} \)
47 \( 1 - 6.44e10T + 5.46e21T^{2} \)
53 \( 1 + 1.40e11T + 2.60e22T^{2} \)
61 \( 1 + 7.41e11T + 1.61e23T^{2} \)
67 \( 1 - 9.59e11T + 5.48e23T^{2} \)
71 \( 1 + 7.60e11T + 1.16e24T^{2} \)
73 \( 1 + 5.28e11T + 1.67e24T^{2} \)
79 \( 1 - 1.53e11T + 4.66e24T^{2} \)
83 \( 1 - 4.32e12T + 8.87e24T^{2} \)
89 \( 1 - 3.47e12T + 2.19e25T^{2} \)
97 \( 1 - 5.67e12T + 6.73e25T^{2} \)
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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

−9.827139377509474236147571009930, −8.868029601771430885704411828441, −7.76402370745592887772949906185, −7.34807607402776588377465423038, −6.03043733699186016684428502865, −4.69666873539473877342769513892, −3.36934912654453739368857648044, −1.99776280762276069655785173422, −0.62114520462066185868108138723, 0, 0.62114520462066185868108138723, 1.99776280762276069655785173422, 3.36934912654453739368857648044, 4.69666873539473877342769513892, 6.03043733699186016684428502865, 7.34807607402776588377465423038, 7.76402370745592887772949906185, 8.868029601771430885704411828441, 9.827139377509474236147571009930

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