Properties

Label 2-177-1.1-c13-0-59
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  + 9.64·2-s + 729·3-s − 8.09e3·4-s − 5.86e4·5-s + 7.02e3·6-s − 6.82e4·7-s − 1.57e5·8-s + 5.31e5·9-s − 5.65e5·10-s − 8.19e6·11-s − 5.90e6·12-s − 4.22e6·13-s − 6.57e5·14-s − 4.27e7·15-s + 6.48e7·16-s + 1.28e8·17-s + 5.12e6·18-s − 5.29e7·19-s + 4.75e8·20-s − 4.97e7·21-s − 7.90e7·22-s − 8.07e7·23-s − 1.14e8·24-s + 2.21e9·25-s − 4.06e7·26-s + 3.87e8·27-s + 5.52e8·28-s + ⋯
L(s)  = 1  + 0.106·2-s + 0.577·3-s − 0.988·4-s − 1.67·5-s + 0.0615·6-s − 0.219·7-s − 0.211·8-s + 0.333·9-s − 0.178·10-s − 1.39·11-s − 0.570·12-s − 0.242·13-s − 0.0233·14-s − 0.969·15-s + 0.966·16-s + 1.28·17-s + 0.0355·18-s − 0.258·19-s + 1.65·20-s − 0.126·21-s − 0.148·22-s − 0.113·23-s − 0.122·24-s + 1.81·25-s − 0.0258·26-s + 0.192·27-s + 0.216·28-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 \( 1 - 9.64T + 8.19e3T^{2} \)
5 \( 1 + 5.86e4T + 1.22e9T^{2} \)
7 \( 1 + 6.82e4T + 9.68e10T^{2} \)
11 \( 1 + 8.19e6T + 3.45e13T^{2} \)
13 \( 1 + 4.22e6T + 3.02e14T^{2} \)
17 \( 1 - 1.28e8T + 9.90e15T^{2} \)
19 \( 1 + 5.29e7T + 4.20e16T^{2} \)
23 \( 1 + 8.07e7T + 5.04e17T^{2} \)
29 \( 1 - 1.70e9T + 1.02e19T^{2} \)
31 \( 1 - 5.22e9T + 2.44e19T^{2} \)
37 \( 1 + 2.22e10T + 2.43e20T^{2} \)
41 \( 1 + 1.90e10T + 9.25e20T^{2} \)
43 \( 1 - 5.00e10T + 1.71e21T^{2} \)
47 \( 1 - 1.40e11T + 5.46e21T^{2} \)
53 \( 1 - 2.75e10T + 2.60e22T^{2} \)
61 \( 1 - 8.31e10T + 1.61e23T^{2} \)
67 \( 1 - 1.44e12T + 5.48e23T^{2} \)
71 \( 1 + 8.00e11T + 1.16e24T^{2} \)
73 \( 1 + 1.15e12T + 1.67e24T^{2} \)
79 \( 1 - 3.68e12T + 4.66e24T^{2} \)
83 \( 1 + 2.67e12T + 8.87e24T^{2} \)
89 \( 1 + 7.91e12T + 2.19e25T^{2} \)
97 \( 1 + 1.18e13T + 6.73e25T^{2} \)
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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

−9.833846069941988687507447496503, −8.513206974349650159245852515768, −8.047818988094380922041559395196, −7.22174300384107666509898422527, −5.42188799599613232723871842871, −4.46530791995538188662146179435, −3.59058457578300348994999727773, −2.78520195304719338440352283427, −0.854880206438068807510300215293, 0, 0.854880206438068807510300215293, 2.78520195304719338440352283427, 3.59058457578300348994999727773, 4.46530791995538188662146179435, 5.42188799599613232723871842871, 7.22174300384107666509898422527, 8.047818988094380922041559395196, 8.513206974349650159245852515768, 9.833846069941988687507447496503

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