Properties

Label 2-177-1.1-c13-0-101
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  − 57.5·2-s − 729·3-s − 4.87e3·4-s + 3.16e4·5-s + 4.19e4·6-s + 2.67e5·7-s + 7.52e5·8-s + 5.31e5·9-s − 1.82e6·10-s + 3.70e6·11-s + 3.55e6·12-s + 3.03e7·13-s − 1.53e7·14-s − 2.30e7·15-s − 3.39e6·16-s + 5.25e6·17-s − 3.06e7·18-s − 1.93e8·19-s − 1.54e8·20-s − 1.94e8·21-s − 2.13e8·22-s − 1.22e8·23-s − 5.48e8·24-s − 2.20e8·25-s − 1.74e9·26-s − 3.87e8·27-s − 1.30e9·28-s + ⋯
L(s)  = 1  − 0.636·2-s − 0.577·3-s − 0.595·4-s + 0.905·5-s + 0.367·6-s + 0.858·7-s + 1.01·8-s + 0.333·9-s − 0.576·10-s + 0.630·11-s + 0.343·12-s + 1.74·13-s − 0.546·14-s − 0.522·15-s − 0.0505·16-s + 0.0527·17-s − 0.212·18-s − 0.943·19-s − 0.538·20-s − 0.495·21-s − 0.401·22-s − 0.172·23-s − 0.585·24-s − 0.180·25-s − 1.11·26-s − 0.192·27-s − 0.510·28-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 \( 1 + 57.5T + 8.19e3T^{2} \)
5 \( 1 - 3.16e4T + 1.22e9T^{2} \)
7 \( 1 - 2.67e5T + 9.68e10T^{2} \)
11 \( 1 - 3.70e6T + 3.45e13T^{2} \)
13 \( 1 - 3.03e7T + 3.02e14T^{2} \)
17 \( 1 - 5.25e6T + 9.90e15T^{2} \)
19 \( 1 + 1.93e8T + 4.20e16T^{2} \)
23 \( 1 + 1.22e8T + 5.04e17T^{2} \)
29 \( 1 - 8.14e8T + 1.02e19T^{2} \)
31 \( 1 - 1.38e9T + 2.44e19T^{2} \)
37 \( 1 + 2.86e10T + 2.43e20T^{2} \)
41 \( 1 + 1.17e10T + 9.25e20T^{2} \)
43 \( 1 + 2.19e10T + 1.71e21T^{2} \)
47 \( 1 + 4.74e10T + 5.46e21T^{2} \)
53 \( 1 + 1.45e11T + 2.60e22T^{2} \)
61 \( 1 + 5.68e10T + 1.61e23T^{2} \)
67 \( 1 - 6.47e11T + 5.48e23T^{2} \)
71 \( 1 + 1.68e11T + 1.16e24T^{2} \)
73 \( 1 + 6.75e11T + 1.67e24T^{2} \)
79 \( 1 + 1.56e11T + 4.66e24T^{2} \)
83 \( 1 + 3.56e12T + 8.87e24T^{2} \)
89 \( 1 + 6.47e12T + 2.19e25T^{2} \)
97 \( 1 + 1.96e12T + 6.73e25T^{2} \)
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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

−9.876236838422829048738129796890, −8.789338796710210637656857027866, −8.212014686572834332685055804553, −6.71026531156852381567719068594, −5.78311635742730341323571311000, −4.76061021039205209746967556746, −3.74090427437022891441746068286, −1.69997021456001406058409535977, −1.31710920969402045741874623875, 0, 1.31710920969402045741874623875, 1.69997021456001406058409535977, 3.74090427437022891441746068286, 4.76061021039205209746967556746, 5.78311635742730341323571311000, 6.71026531156852381567719068594, 8.212014686572834332685055804553, 8.789338796710210637656857027866, 9.876236838422829048738129796890

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