Properties

Label 2-177-1.1-c11-0-83
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 47.2·2-s + 243·3-s + 189.·4-s + 4.20e3·5-s − 1.14e4·6-s + 8.23e3·7-s + 8.79e4·8-s + 5.90e4·9-s − 1.98e5·10-s + 2.81e5·11-s + 4.59e4·12-s − 2.86e5·13-s − 3.89e5·14-s + 1.02e6·15-s − 4.54e6·16-s + 5.87e6·17-s − 2.79e6·18-s − 8.24e6·19-s + 7.95e5·20-s + 2.00e6·21-s − 1.33e7·22-s − 1.83e7·23-s + 2.13e7·24-s − 3.11e7·25-s + 1.35e7·26-s + 1.43e7·27-s + 1.55e6·28-s + ⋯
L(s)  = 1  − 1.04·2-s + 0.577·3-s + 0.0923·4-s + 0.601·5-s − 0.603·6-s + 0.185·7-s + 0.948·8-s + 0.333·9-s − 0.628·10-s + 0.527·11-s + 0.0533·12-s − 0.213·13-s − 0.193·14-s + 0.347·15-s − 1.08·16-s + 1.00·17-s − 0.348·18-s − 0.763·19-s + 0.0555·20-s + 0.106·21-s − 0.551·22-s − 0.594·23-s + 0.547·24-s − 0.637·25-s + 0.223·26-s + 0.192·27-s + 0.0171·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 47.2T + 2.04e3T^{2} \)
5 \( 1 - 4.20e3T + 4.88e7T^{2} \)
7 \( 1 - 8.23e3T + 1.97e9T^{2} \)
11 \( 1 - 2.81e5T + 2.85e11T^{2} \)
13 \( 1 + 2.86e5T + 1.79e12T^{2} \)
17 \( 1 - 5.87e6T + 3.42e13T^{2} \)
19 \( 1 + 8.24e6T + 1.16e14T^{2} \)
23 \( 1 + 1.83e7T + 9.52e14T^{2} \)
29 \( 1 + 3.47e7T + 1.22e16T^{2} \)
31 \( 1 - 3.33e7T + 2.54e16T^{2} \)
37 \( 1 + 6.02e8T + 1.77e17T^{2} \)
41 \( 1 - 1.14e9T + 5.50e17T^{2} \)
43 \( 1 + 9.42e8T + 9.29e17T^{2} \)
47 \( 1 + 1.01e9T + 2.47e18T^{2} \)
53 \( 1 + 1.59e9T + 9.26e18T^{2} \)
61 \( 1 + 5.49e9T + 4.35e19T^{2} \)
67 \( 1 + 9.28e9T + 1.22e20T^{2} \)
71 \( 1 - 5.67e9T + 2.31e20T^{2} \)
73 \( 1 + 3.15e9T + 3.13e20T^{2} \)
79 \( 1 - 3.09e10T + 7.47e20T^{2} \)
83 \( 1 - 6.71e9T + 1.28e21T^{2} \)
89 \( 1 - 8.36e10T + 2.77e21T^{2} \)
97 \( 1 - 1.29e11T + 7.15e21T^{2} \)
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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

−9.874117448557412506893481832994, −9.243575599760181596531279497244, −8.282598649951772188856138280615, −7.51235820406061292979456607595, −6.24000163217656322559241339523, −4.84523159091700507924317119431, −3.62514137547653884198230650210, −2.08454272287988560391839945489, −1.32872290754201231267139350424, 0, 1.32872290754201231267139350424, 2.08454272287988560391839945489, 3.62514137547653884198230650210, 4.84523159091700507924317119431, 6.24000163217656322559241339523, 7.51235820406061292979456607595, 8.282598649951772188856138280615, 9.243575599760181596531279497244, 9.874117448557412506893481832994

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