Properties

Label 2-177-1.1-c13-0-87
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  − 73.9·2-s + 729·3-s − 2.72e3·4-s − 2.27e4·5-s − 5.38e4·6-s + 4.05e5·7-s + 8.07e5·8-s + 5.31e5·9-s + 1.67e6·10-s − 6.65e6·11-s − 1.98e6·12-s + 2.06e7·13-s − 2.99e7·14-s − 1.65e7·15-s − 3.73e7·16-s + 2.16e7·17-s − 3.92e7·18-s + 1.30e8·19-s + 6.19e7·20-s + 2.95e8·21-s + 4.91e8·22-s − 7.96e8·23-s + 5.88e8·24-s − 7.04e8·25-s − 1.52e9·26-s + 3.87e8·27-s − 1.10e9·28-s + ⋯
L(s)  = 1  − 0.816·2-s + 0.577·3-s − 0.332·4-s − 0.650·5-s − 0.471·6-s + 1.30·7-s + 1.08·8-s + 0.333·9-s + 0.531·10-s − 1.13·11-s − 0.192·12-s + 1.18·13-s − 1.06·14-s − 0.375·15-s − 0.556·16-s + 0.217·17-s − 0.272·18-s + 0.636·19-s + 0.216·20-s + 0.752·21-s + 0.924·22-s − 1.12·23-s + 0.628·24-s − 0.577·25-s − 0.969·26-s + 0.192·27-s − 0.433·28-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 \( 1 + 73.9T + 8.19e3T^{2} \)
5 \( 1 + 2.27e4T + 1.22e9T^{2} \)
7 \( 1 - 4.05e5T + 9.68e10T^{2} \)
11 \( 1 + 6.65e6T + 3.45e13T^{2} \)
13 \( 1 - 2.06e7T + 3.02e14T^{2} \)
17 \( 1 - 2.16e7T + 9.90e15T^{2} \)
19 \( 1 - 1.30e8T + 4.20e16T^{2} \)
23 \( 1 + 7.96e8T + 5.04e17T^{2} \)
29 \( 1 + 3.04e9T + 1.02e19T^{2} \)
31 \( 1 + 2.02e9T + 2.44e19T^{2} \)
37 \( 1 - 5.59e9T + 2.43e20T^{2} \)
41 \( 1 - 3.73e9T + 9.25e20T^{2} \)
43 \( 1 - 6.33e10T + 1.71e21T^{2} \)
47 \( 1 + 2.15e10T + 5.46e21T^{2} \)
53 \( 1 + 2.07e11T + 2.60e22T^{2} \)
61 \( 1 + 1.72e11T + 1.61e23T^{2} \)
67 \( 1 - 6.74e11T + 5.48e23T^{2} \)
71 \( 1 - 3.55e10T + 1.16e24T^{2} \)
73 \( 1 - 2.22e11T + 1.67e24T^{2} \)
79 \( 1 + 5.78e11T + 4.66e24T^{2} \)
83 \( 1 + 3.89e12T + 8.87e24T^{2} \)
89 \( 1 - 5.37e12T + 2.19e25T^{2} \)
97 \( 1 + 7.20e12T + 6.73e25T^{2} \)
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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

−9.690253525155145120768648416076, −8.647370809591919008685742763683, −7.85516327048820807038147844652, −7.64262572612860036365654743667, −5.62557839453090949270334521647, −4.50739994473868002879474878908, −3.62157270383032768726233181288, −2.05666852442161990038279045567, −1.14375573083725175118623114777, 0, 1.14375573083725175118623114777, 2.05666852442161990038279045567, 3.62157270383032768726233181288, 4.50739994473868002879474878908, 5.62557839453090949270334521647, 7.64262572612860036365654743667, 7.85516327048820807038147844652, 8.647370809591919008685742763683, 9.690253525155145120768648416076

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