Properties

Label 2-177-1.1-c13-0-103
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  + 105.·2-s + 729·3-s + 2.85e3·4-s − 5.72e4·5-s + 7.66e4·6-s + 1.67e5·7-s − 5.60e5·8-s + 5.31e5·9-s − 6.02e6·10-s + 7.18e6·11-s + 2.08e6·12-s − 1.39e7·13-s + 1.75e7·14-s − 4.17e7·15-s − 8.23e7·16-s + 8.87e7·17-s + 5.58e7·18-s + 2.60e8·19-s − 1.63e8·20-s + 1.21e8·21-s + 7.54e8·22-s − 4.15e6·23-s − 4.08e8·24-s + 2.06e9·25-s − 1.46e9·26-s + 3.87e8·27-s + 4.76e8·28-s + ⋯
L(s)  = 1  + 1.16·2-s + 0.577·3-s + 0.348·4-s − 1.63·5-s + 0.670·6-s + 0.536·7-s − 0.756·8-s + 0.333·9-s − 1.90·10-s + 1.22·11-s + 0.201·12-s − 0.802·13-s + 0.623·14-s − 0.946·15-s − 1.22·16-s + 0.891·17-s + 0.387·18-s + 1.27·19-s − 0.571·20-s + 0.309·21-s + 1.41·22-s − 0.00585·23-s − 0.436·24-s + 1.68·25-s − 0.931·26-s + 0.192·27-s + 0.187·28-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 \( 1 - 105.T + 8.19e3T^{2} \)
5 \( 1 + 5.72e4T + 1.22e9T^{2} \)
7 \( 1 - 1.67e5T + 9.68e10T^{2} \)
11 \( 1 - 7.18e6T + 3.45e13T^{2} \)
13 \( 1 + 1.39e7T + 3.02e14T^{2} \)
17 \( 1 - 8.87e7T + 9.90e15T^{2} \)
19 \( 1 - 2.60e8T + 4.20e16T^{2} \)
23 \( 1 + 4.15e6T + 5.04e17T^{2} \)
29 \( 1 + 2.99e9T + 1.02e19T^{2} \)
31 \( 1 - 3.28e9T + 2.44e19T^{2} \)
37 \( 1 + 1.78e10T + 2.43e20T^{2} \)
41 \( 1 - 2.19e10T + 9.25e20T^{2} \)
43 \( 1 + 4.99e10T + 1.71e21T^{2} \)
47 \( 1 + 2.39e10T + 5.46e21T^{2} \)
53 \( 1 - 1.15e11T + 2.60e22T^{2} \)
61 \( 1 + 2.14e11T + 1.61e23T^{2} \)
67 \( 1 + 3.42e11T + 5.48e23T^{2} \)
71 \( 1 + 2.03e11T + 1.16e24T^{2} \)
73 \( 1 - 6.31e11T + 1.67e24T^{2} \)
79 \( 1 + 2.19e12T + 4.66e24T^{2} \)
83 \( 1 + 3.95e12T + 8.87e24T^{2} \)
89 \( 1 + 7.60e12T + 2.19e25T^{2} \)
97 \( 1 - 2.19e11T + 6.73e25T^{2} \)
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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

−9.741022877606605532601447516277, −8.687523732049830857430830150431, −7.72191131720089974072177108273, −6.90755139212671329484682838107, −5.34586104455878845338219260253, −4.41252727285526125167628911400, −3.67930751430763033625519903078, −2.98751218930527425306764878755, −1.31888874959207055947505698860, 0, 1.31888874959207055947505698860, 2.98751218930527425306764878755, 3.67930751430763033625519903078, 4.41252727285526125167628911400, 5.34586104455878845338219260253, 6.90755139212671329484682838107, 7.72191131720089974072177108273, 8.687523732049830857430830150431, 9.741022877606605532601447516277

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