Properties

Label 2-177-1.1-c13-0-100
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  − 44.2·2-s + 729·3-s − 6.23e3·4-s + 1.48e4·5-s − 3.22e4·6-s + 4.73e5·7-s + 6.38e5·8-s + 5.31e5·9-s − 6.58e5·10-s − 5.62e6·11-s − 4.54e6·12-s − 1.37e7·13-s − 2.09e7·14-s + 1.08e7·15-s + 2.28e7·16-s + 1.90e8·17-s − 2.34e7·18-s − 2.39e8·19-s − 9.29e7·20-s + 3.45e8·21-s + 2.48e8·22-s + 8.48e8·23-s + 4.65e8·24-s − 9.98e8·25-s + 6.07e8·26-s + 3.87e8·27-s − 2.95e9·28-s + ⋯
L(s)  = 1  − 0.488·2-s + 0.577·3-s − 0.761·4-s + 0.426·5-s − 0.282·6-s + 1.52·7-s + 0.860·8-s + 0.333·9-s − 0.208·10-s − 0.957·11-s − 0.439·12-s − 0.790·13-s − 0.743·14-s + 0.246·15-s + 0.340·16-s + 1.91·17-s − 0.162·18-s − 1.16·19-s − 0.324·20-s + 0.878·21-s + 0.467·22-s + 1.19·23-s + 0.496·24-s − 0.818·25-s + 0.385·26-s + 0.192·27-s − 1.15·28-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 \( 1 + 44.2T + 8.19e3T^{2} \)
5 \( 1 - 1.48e4T + 1.22e9T^{2} \)
7 \( 1 - 4.73e5T + 9.68e10T^{2} \)
11 \( 1 + 5.62e6T + 3.45e13T^{2} \)
13 \( 1 + 1.37e7T + 3.02e14T^{2} \)
17 \( 1 - 1.90e8T + 9.90e15T^{2} \)
19 \( 1 + 2.39e8T + 4.20e16T^{2} \)
23 \( 1 - 8.48e8T + 5.04e17T^{2} \)
29 \( 1 + 2.26e9T + 1.02e19T^{2} \)
31 \( 1 - 6.87e8T + 2.44e19T^{2} \)
37 \( 1 + 1.85e10T + 2.43e20T^{2} \)
41 \( 1 + 2.65e10T + 9.25e20T^{2} \)
43 \( 1 + 8.00e10T + 1.71e21T^{2} \)
47 \( 1 + 1.28e11T + 5.46e21T^{2} \)
53 \( 1 - 2.46e11T + 2.60e22T^{2} \)
61 \( 1 - 6.06e11T + 1.61e23T^{2} \)
67 \( 1 - 1.30e11T + 5.48e23T^{2} \)
71 \( 1 + 1.05e12T + 1.16e24T^{2} \)
73 \( 1 + 1.45e12T + 1.67e24T^{2} \)
79 \( 1 - 9.47e10T + 4.66e24T^{2} \)
83 \( 1 - 4.49e12T + 8.87e24T^{2} \)
89 \( 1 - 4.97e11T + 2.19e25T^{2} \)
97 \( 1 + 1.63e12T + 6.73e25T^{2} \)
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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

−9.923943365521574502936251444766, −8.649653181404452606741608889820, −8.084299906251712106630025922080, −7.30528934689016606231398548686, −5.31306196330812773647539240917, −4.89530973486165262183017417365, −3.51141720442754803145178748802, −2.08272454709999862387050883480, −1.31662365291902374588883195624, 0, 1.31662365291902374588883195624, 2.08272454709999862387050883480, 3.51141720442754803145178748802, 4.89530973486165262183017417365, 5.31306196330812773647539240917, 7.30528934689016606231398548686, 8.084299906251712106630025922080, 8.649653181404452606741608889820, 9.923943365521574502936251444766

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