Properties

Label 2-177-1.1-c13-0-82
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  + 20.8·2-s − 729·3-s − 7.75e3·4-s + 2.78e4·5-s − 1.52e4·6-s + 6.11e4·7-s − 3.33e5·8-s + 5.31e5·9-s + 5.81e5·10-s − 3.53e6·11-s + 5.65e6·12-s + 2.05e7·13-s + 1.27e6·14-s − 2.03e7·15-s + 5.65e7·16-s − 1.17e8·17-s + 1.10e7·18-s + 3.76e6·19-s − 2.16e8·20-s − 4.45e7·21-s − 7.37e7·22-s − 6.80e8·23-s + 2.42e8·24-s − 4.44e8·25-s + 4.28e8·26-s − 3.87e8·27-s − 4.74e8·28-s + ⋯
L(s)  = 1  + 0.230·2-s − 0.577·3-s − 0.946·4-s + 0.797·5-s − 0.133·6-s + 0.196·7-s − 0.449·8-s + 0.333·9-s + 0.184·10-s − 0.600·11-s + 0.546·12-s + 1.18·13-s + 0.0452·14-s − 0.460·15-s + 0.843·16-s − 1.17·17-s + 0.0769·18-s + 0.0183·19-s − 0.755·20-s − 0.113·21-s − 0.138·22-s − 0.958·23-s + 0.259·24-s − 0.363·25-s + 0.272·26-s − 0.192·27-s − 0.185·28-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 \( 1 - 20.8T + 8.19e3T^{2} \)
5 \( 1 - 2.78e4T + 1.22e9T^{2} \)
7 \( 1 - 6.11e4T + 9.68e10T^{2} \)
11 \( 1 + 3.53e6T + 3.45e13T^{2} \)
13 \( 1 - 2.05e7T + 3.02e14T^{2} \)
17 \( 1 + 1.17e8T + 9.90e15T^{2} \)
19 \( 1 - 3.76e6T + 4.20e16T^{2} \)
23 \( 1 + 6.80e8T + 5.04e17T^{2} \)
29 \( 1 - 3.42e9T + 1.02e19T^{2} \)
31 \( 1 + 1.83e9T + 2.44e19T^{2} \)
37 \( 1 - 1.49e10T + 2.43e20T^{2} \)
41 \( 1 - 2.74e10T + 9.25e20T^{2} \)
43 \( 1 + 4.39e10T + 1.71e21T^{2} \)
47 \( 1 - 4.98e10T + 5.46e21T^{2} \)
53 \( 1 - 1.99e11T + 2.60e22T^{2} \)
61 \( 1 - 2.57e11T + 1.61e23T^{2} \)
67 \( 1 - 1.02e11T + 5.48e23T^{2} \)
71 \( 1 + 1.55e12T + 1.16e24T^{2} \)
73 \( 1 - 2.32e12T + 1.67e24T^{2} \)
79 \( 1 + 3.36e11T + 4.66e24T^{2} \)
83 \( 1 - 1.25e12T + 8.87e24T^{2} \)
89 \( 1 - 4.64e12T + 2.19e25T^{2} \)
97 \( 1 - 1.27e12T + 6.73e25T^{2} \)
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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

−9.900373747999741707149457762602, −8.917139468251612789379859703717, −7.998928791641881348446063400214, −6.40758234803769771123249759244, −5.72218052456918534327490828980, −4.74535456453310017565051854369, −3.80918994800569713022958400973, −2.30272003592800813924981849137, −1.07516746714422749031090265999, 0, 1.07516746714422749031090265999, 2.30272003592800813924981849137, 3.80918994800569713022958400973, 4.74535456453310017565051854369, 5.72218052456918534327490828980, 6.40758234803769771123249759244, 7.998928791641881348446063400214, 8.917139468251612789379859703717, 9.900373747999741707149457762602

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