Properties

Label 2-177-1.1-c13-0-115
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  + 132.·2-s + 729·3-s + 9.44e3·4-s + 3.27e3·5-s + 9.68e4·6-s − 2.96e5·7-s + 1.66e5·8-s + 5.31e5·9-s + 4.35e5·10-s + 9.14e6·11-s + 6.88e6·12-s − 3.48e6·13-s − 3.93e7·14-s + 2.38e6·15-s − 5.52e7·16-s − 3.85e7·17-s + 7.05e7·18-s + 2.85e7·19-s + 3.09e7·20-s − 2.16e8·21-s + 1.21e9·22-s − 7.24e8·23-s + 1.21e8·24-s − 1.20e9·25-s − 4.62e8·26-s + 3.87e8·27-s − 2.79e9·28-s + ⋯
L(s)  = 1  + 1.46·2-s + 0.577·3-s + 1.15·4-s + 0.0938·5-s + 0.847·6-s − 0.952·7-s + 0.224·8-s + 0.333·9-s + 0.137·10-s + 1.55·11-s + 0.665·12-s − 0.200·13-s − 1.39·14-s + 0.0541·15-s − 0.823·16-s − 0.387·17-s + 0.489·18-s + 0.139·19-s + 0.108·20-s − 0.549·21-s + 2.28·22-s − 1.02·23-s + 0.129·24-s − 0.991·25-s − 0.293·26-s + 0.192·27-s − 1.09·28-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 \( 1 - 132.T + 8.19e3T^{2} \)
5 \( 1 - 3.27e3T + 1.22e9T^{2} \)
7 \( 1 + 2.96e5T + 9.68e10T^{2} \)
11 \( 1 - 9.14e6T + 3.45e13T^{2} \)
13 \( 1 + 3.48e6T + 3.02e14T^{2} \)
17 \( 1 + 3.85e7T + 9.90e15T^{2} \)
19 \( 1 - 2.85e7T + 4.20e16T^{2} \)
23 \( 1 + 7.24e8T + 5.04e17T^{2} \)
29 \( 1 + 3.17e9T + 1.02e19T^{2} \)
31 \( 1 + 4.23e9T + 2.44e19T^{2} \)
37 \( 1 - 2.66e10T + 2.43e20T^{2} \)
41 \( 1 + 2.67e10T + 9.25e20T^{2} \)
43 \( 1 - 7.18e10T + 1.71e21T^{2} \)
47 \( 1 - 7.60e10T + 5.46e21T^{2} \)
53 \( 1 + 2.54e11T + 2.60e22T^{2} \)
61 \( 1 + 4.30e11T + 1.61e23T^{2} \)
67 \( 1 + 1.20e12T + 5.48e23T^{2} \)
71 \( 1 - 6.68e11T + 1.16e24T^{2} \)
73 \( 1 + 2.42e11T + 1.67e24T^{2} \)
79 \( 1 - 4.77e11T + 4.66e24T^{2} \)
83 \( 1 + 2.06e12T + 8.87e24T^{2} \)
89 \( 1 + 6.68e12T + 2.19e25T^{2} \)
97 \( 1 - 3.13e12T + 6.73e25T^{2} \)
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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

−9.624795303704834871049694384962, −9.116792647453795325191276296048, −7.51746043777932184950740313155, −6.43119854785977762511608059521, −5.81242882170050082405345212737, −4.25987209928981656181817720411, −3.78293370735865742124461846954, −2.76437868733829708114879129883, −1.67861236915294138003054145349, 0, 1.67861236915294138003054145349, 2.76437868733829708114879129883, 3.78293370735865742124461846954, 4.25987209928981656181817720411, 5.81242882170050082405345212737, 6.43119854785977762511608059521, 7.51746043777932184950740313155, 9.116792647453795325191276296048, 9.624795303704834871049694384962

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