Properties

Label 2-177-1.1-c13-0-99
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  − 150.·2-s + 729·3-s + 1.45e4·4-s + 4.04e4·5-s − 1.09e5·6-s − 1.11e5·7-s − 9.53e5·8-s + 5.31e5·9-s − 6.08e6·10-s + 1.14e7·11-s + 1.05e7·12-s − 2.30e7·13-s + 1.68e7·14-s + 2.94e7·15-s + 2.48e7·16-s − 3.16e7·17-s − 8.00e7·18-s − 1.79e8·19-s + 5.86e8·20-s − 8.14e7·21-s − 1.72e9·22-s + 2.82e8·23-s − 6.95e8·24-s + 4.11e8·25-s + 3.47e9·26-s + 3.87e8·27-s − 1.62e9·28-s + ⋯
L(s)  = 1  − 1.66·2-s + 0.577·3-s + 1.77·4-s + 1.15·5-s − 0.961·6-s − 0.358·7-s − 1.28·8-s + 0.333·9-s − 1.92·10-s + 1.94·11-s + 1.02·12-s − 1.32·13-s + 0.597·14-s + 0.667·15-s + 0.369·16-s − 0.317·17-s − 0.555·18-s − 0.873·19-s + 2.04·20-s − 0.207·21-s − 3.24·22-s + 0.398·23-s − 0.742·24-s + 0.337·25-s + 2.20·26-s + 0.192·27-s − 0.636·28-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 \( 1 + 150.T + 8.19e3T^{2} \)
5 \( 1 - 4.04e4T + 1.22e9T^{2} \)
7 \( 1 + 1.11e5T + 9.68e10T^{2} \)
11 \( 1 - 1.14e7T + 3.45e13T^{2} \)
13 \( 1 + 2.30e7T + 3.02e14T^{2} \)
17 \( 1 + 3.16e7T + 9.90e15T^{2} \)
19 \( 1 + 1.79e8T + 4.20e16T^{2} \)
23 \( 1 - 2.82e8T + 5.04e17T^{2} \)
29 \( 1 - 4.89e9T + 1.02e19T^{2} \)
31 \( 1 - 6.10e9T + 2.44e19T^{2} \)
37 \( 1 + 2.53e10T + 2.43e20T^{2} \)
41 \( 1 + 8.94e8T + 9.25e20T^{2} \)
43 \( 1 + 5.19e10T + 1.71e21T^{2} \)
47 \( 1 + 8.00e10T + 5.46e21T^{2} \)
53 \( 1 + 3.21e11T + 2.60e22T^{2} \)
61 \( 1 - 9.93e10T + 1.61e23T^{2} \)
67 \( 1 + 3.29e11T + 5.48e23T^{2} \)
71 \( 1 - 1.94e12T + 1.16e24T^{2} \)
73 \( 1 + 4.35e11T + 1.67e24T^{2} \)
79 \( 1 + 3.67e12T + 4.66e24T^{2} \)
83 \( 1 + 3.03e12T + 8.87e24T^{2} \)
89 \( 1 + 6.66e11T + 2.19e25T^{2} \)
97 \( 1 - 1.40e12T + 6.73e25T^{2} \)
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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

−9.771924207537815236932726827944, −9.002095727988703303758216824875, −8.255936576746699927501555367173, −6.71027426951721583960686427745, −6.57814871627938887352283146950, −4.63923800147198252960564130795, −2.97744511757422615228286803821, −1.92700950985762750549131514732, −1.31358346268348188306697624431, 0, 1.31358346268348188306697624431, 1.92700950985762750549131514732, 2.97744511757422615228286803821, 4.63923800147198252960564130795, 6.57814871627938887352283146950, 6.71027426951721583960686427745, 8.255936576746699927501555367173, 9.002095727988703303758216824875, 9.771924207537815236932726827944

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