Properties

Label 2-177-1.1-c11-0-23
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 66.1·2-s − 243·3-s + 2.32e3·4-s + 6.88e3·5-s + 1.60e4·6-s + 7.66e3·7-s − 1.83e4·8-s + 5.90e4·9-s − 4.55e5·10-s + 2.65e5·11-s − 5.65e5·12-s − 2.03e6·13-s − 5.07e5·14-s − 1.67e6·15-s − 3.54e6·16-s − 2.19e6·17-s − 3.90e6·18-s + 1.39e7·19-s + 1.60e7·20-s − 1.86e6·21-s − 1.75e7·22-s − 1.47e7·23-s + 4.45e6·24-s − 1.45e6·25-s + 1.34e8·26-s − 1.43e7·27-s + 1.78e7·28-s + ⋯
L(s)  = 1  − 1.46·2-s − 0.577·3-s + 1.13·4-s + 0.984·5-s + 0.843·6-s + 0.172·7-s − 0.197·8-s + 0.333·9-s − 1.43·10-s + 0.496·11-s − 0.655·12-s − 1.51·13-s − 0.252·14-s − 0.568·15-s − 0.846·16-s − 0.374·17-s − 0.487·18-s + 1.28·19-s + 1.11·20-s − 0.0995·21-s − 0.725·22-s − 0.479·23-s + 0.114·24-s − 0.0298·25-s + 2.21·26-s − 0.192·27-s + 0.195·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.8813260428\)
\(L(\frac12)\) \(\approx\) \(0.8813260428\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 + 7.14e8T \)
good2 \( 1 + 66.1T + 2.04e3T^{2} \)
5 \( 1 - 6.88e3T + 4.88e7T^{2} \)
7 \( 1 - 7.66e3T + 1.97e9T^{2} \)
11 \( 1 - 2.65e5T + 2.85e11T^{2} \)
13 \( 1 + 2.03e6T + 1.79e12T^{2} \)
17 \( 1 + 2.19e6T + 3.42e13T^{2} \)
19 \( 1 - 1.39e7T + 1.16e14T^{2} \)
23 \( 1 + 1.47e7T + 9.52e14T^{2} \)
29 \( 1 - 1.32e8T + 1.22e16T^{2} \)
31 \( 1 + 6.41e7T + 2.54e16T^{2} \)
37 \( 1 - 2.46e8T + 1.77e17T^{2} \)
41 \( 1 + 1.45e7T + 5.50e17T^{2} \)
43 \( 1 - 6.41e7T + 9.29e17T^{2} \)
47 \( 1 - 1.68e9T + 2.47e18T^{2} \)
53 \( 1 - 5.02e9T + 9.26e18T^{2} \)
61 \( 1 - 1.12e9T + 4.35e19T^{2} \)
67 \( 1 + 1.51e10T + 1.22e20T^{2} \)
71 \( 1 + 5.00e9T + 2.31e20T^{2} \)
73 \( 1 - 2.20e10T + 3.13e20T^{2} \)
79 \( 1 + 4.29e10T + 7.47e20T^{2} \)
83 \( 1 + 1.16e10T + 1.28e21T^{2} \)
89 \( 1 + 8.90e10T + 2.77e21T^{2} \)
97 \( 1 - 1.57e11T + 7.15e21T^{2} \)
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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

−10.19480682445242301044682197571, −9.792815107475663927970164563103, −8.934207013182390568754930228834, −7.65267168603556492034918004016, −6.85523169894088210561065736397, −5.68231265988282911924609529886, −4.55413462765532997769897417835, −2.52606564274076224758253131318, −1.55530833967196743140051222736, −0.57198343367832473705210833368, 0.57198343367832473705210833368, 1.55530833967196743140051222736, 2.52606564274076224758253131318, 4.55413462765532997769897417835, 5.68231265988282911924609529886, 6.85523169894088210561065736397, 7.65267168603556492034918004016, 8.934207013182390568754930228834, 9.792815107475663927970164563103, 10.19480682445242301044682197571

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