Properties

Label 2-177-1.1-c13-0-57
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  − 141.·2-s + 729·3-s + 1.16e4·4-s − 3.25e4·5-s − 1.02e5·6-s − 3.39e5·7-s − 4.93e5·8-s + 5.31e5·9-s + 4.58e6·10-s + 3.57e6·11-s + 8.52e6·12-s − 2.09e7·13-s + 4.78e7·14-s − 2.37e7·15-s − 2.61e7·16-s + 9.09e6·17-s − 7.49e7·18-s + 1.92e8·19-s − 3.80e8·20-s − 2.47e8·21-s − 5.03e8·22-s − 8.21e8·23-s − 3.59e8·24-s − 1.61e8·25-s + 2.94e9·26-s + 3.87e8·27-s − 3.96e9·28-s + ⋯
L(s)  = 1  − 1.55·2-s + 0.577·3-s + 1.42·4-s − 0.931·5-s − 0.899·6-s − 1.09·7-s − 0.665·8-s + 0.333·9-s + 1.45·10-s + 0.608·11-s + 0.823·12-s − 1.20·13-s + 1.69·14-s − 0.537·15-s − 0.390·16-s + 0.0913·17-s − 0.519·18-s + 0.939·19-s − 1.32·20-s − 0.629·21-s − 0.947·22-s − 1.15·23-s − 0.384·24-s − 0.132·25-s + 1.87·26-s + 0.192·27-s − 1.55·28-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 \( 1 + 141.T + 8.19e3T^{2} \)
5 \( 1 + 3.25e4T + 1.22e9T^{2} \)
7 \( 1 + 3.39e5T + 9.68e10T^{2} \)
11 \( 1 - 3.57e6T + 3.45e13T^{2} \)
13 \( 1 + 2.09e7T + 3.02e14T^{2} \)
17 \( 1 - 9.09e6T + 9.90e15T^{2} \)
19 \( 1 - 1.92e8T + 4.20e16T^{2} \)
23 \( 1 + 8.21e8T + 5.04e17T^{2} \)
29 \( 1 + 5.08e8T + 1.02e19T^{2} \)
31 \( 1 - 4.32e9T + 2.44e19T^{2} \)
37 \( 1 + 6.74e9T + 2.43e20T^{2} \)
41 \( 1 - 4.16e10T + 9.25e20T^{2} \)
43 \( 1 - 6.89e10T + 1.71e21T^{2} \)
47 \( 1 + 6.47e10T + 5.46e21T^{2} \)
53 \( 1 - 8.27e10T + 2.60e22T^{2} \)
61 \( 1 + 3.81e11T + 1.61e23T^{2} \)
67 \( 1 + 8.52e11T + 5.48e23T^{2} \)
71 \( 1 + 1.49e12T + 1.16e24T^{2} \)
73 \( 1 - 1.20e11T + 1.67e24T^{2} \)
79 \( 1 - 3.34e12T + 4.66e24T^{2} \)
83 \( 1 - 2.74e12T + 8.87e24T^{2} \)
89 \( 1 - 5.23e12T + 2.19e25T^{2} \)
97 \( 1 - 9.88e12T + 6.73e25T^{2} \)
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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

−9.615966412440685357975892928396, −9.034313862650941375942842530967, −7.78703224349688578560583001306, −7.42823580038704959653321674424, −6.27414121052474330364617734667, −4.37400160494691432149976663393, −3.24893305128972194273037587448, −2.18290756961911853571049511673, −0.842467303272546529040459571874, 0, 0.842467303272546529040459571874, 2.18290756961911853571049511673, 3.24893305128972194273037587448, 4.37400160494691432149976663393, 6.27414121052474330364617734667, 7.42823580038704959653321674424, 7.78703224349688578560583001306, 9.034313862650941375942842530967, 9.615966412440685357975892928396

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