Properties

Label 2-177-1.1-c13-0-95
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  + 20.2·2-s + 729·3-s − 7.78e3·4-s + 3.72e4·5-s + 1.47e4·6-s − 2.60e5·7-s − 3.23e5·8-s + 5.31e5·9-s + 7.54e5·10-s − 3.65e6·11-s − 5.67e6·12-s − 1.12e7·13-s − 5.27e6·14-s + 2.71e7·15-s + 5.71e7·16-s + 1.19e6·17-s + 1.07e7·18-s + 2.27e8·19-s − 2.89e8·20-s − 1.89e8·21-s − 7.40e7·22-s + 8.27e8·23-s − 2.36e8·24-s + 1.65e8·25-s − 2.28e8·26-s + 3.87e8·27-s + 2.02e9·28-s + ⋯
L(s)  = 1  + 0.223·2-s + 0.577·3-s − 0.949·4-s + 1.06·5-s + 0.129·6-s − 0.835·7-s − 0.436·8-s + 0.333·9-s + 0.238·10-s − 0.621·11-s − 0.548·12-s − 0.647·13-s − 0.187·14-s + 0.615·15-s + 0.852·16-s + 0.0120·17-s + 0.0746·18-s + 1.10·19-s − 1.01·20-s − 0.482·21-s − 0.139·22-s + 1.16·23-s − 0.252·24-s + 0.135·25-s − 0.145·26-s + 0.192·27-s + 0.794·28-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 \( 1 - 20.2T + 8.19e3T^{2} \)
5 \( 1 - 3.72e4T + 1.22e9T^{2} \)
7 \( 1 + 2.60e5T + 9.68e10T^{2} \)
11 \( 1 + 3.65e6T + 3.45e13T^{2} \)
13 \( 1 + 1.12e7T + 3.02e14T^{2} \)
17 \( 1 - 1.19e6T + 9.90e15T^{2} \)
19 \( 1 - 2.27e8T + 4.20e16T^{2} \)
23 \( 1 - 8.27e8T + 5.04e17T^{2} \)
29 \( 1 - 1.28e9T + 1.02e19T^{2} \)
31 \( 1 - 3.95e9T + 2.44e19T^{2} \)
37 \( 1 + 9.79e9T + 2.43e20T^{2} \)
41 \( 1 + 3.95e10T + 9.25e20T^{2} \)
43 \( 1 - 6.25e10T + 1.71e21T^{2} \)
47 \( 1 + 8.64e10T + 5.46e21T^{2} \)
53 \( 1 + 6.34e10T + 2.60e22T^{2} \)
61 \( 1 - 2.34e11T + 1.61e23T^{2} \)
67 \( 1 + 1.03e12T + 5.48e23T^{2} \)
71 \( 1 + 1.82e12T + 1.16e24T^{2} \)
73 \( 1 - 2.32e12T + 1.67e24T^{2} \)
79 \( 1 + 8.66e11T + 4.66e24T^{2} \)
83 \( 1 - 4.79e12T + 8.87e24T^{2} \)
89 \( 1 + 1.11e12T + 2.19e25T^{2} \)
97 \( 1 + 5.79e12T + 6.73e25T^{2} \)
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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

−9.677094075065509742817286726443, −9.139889323112457045394193152241, −7.973851510380976543349832948913, −6.73731536042881440199782261188, −5.55282529442105532471757292435, −4.77837547927584047848389327568, −3.35450222698237068632568795787, −2.63572909884115572016938983702, −1.22972387435969187277819529747, 0, 1.22972387435969187277819529747, 2.63572909884115572016938983702, 3.35450222698237068632568795787, 4.77837547927584047848389327568, 5.55282529442105532471757292435, 6.73731536042881440199782261188, 7.973851510380976543349832948913, 9.139889323112457045394193152241, 9.677094075065509742817286726443

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