Properties

Label 2-177-1.1-c13-0-79
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  − 158.·2-s + 729·3-s + 1.67e4·4-s − 6.19e4·5-s − 1.15e5·6-s + 1.40e5·7-s − 1.35e6·8-s + 5.31e5·9-s + 9.78e6·10-s + 1.00e7·11-s + 1.22e7·12-s + 3.21e7·13-s − 2.21e7·14-s − 4.51e7·15-s + 7.71e7·16-s − 1.32e8·17-s − 8.39e7·18-s + 8.81e7·19-s − 1.03e9·20-s + 1.02e8·21-s − 1.59e9·22-s − 9.30e8·23-s − 9.90e8·24-s + 2.61e9·25-s − 5.07e9·26-s + 3.87e8·27-s + 2.35e9·28-s + ⋯
L(s)  = 1  − 1.74·2-s + 0.577·3-s + 2.04·4-s − 1.77·5-s − 1.00·6-s + 0.449·7-s − 1.83·8-s + 0.333·9-s + 3.09·10-s + 1.71·11-s + 1.18·12-s + 1.84·13-s − 0.785·14-s − 1.02·15-s + 1.15·16-s − 1.33·17-s − 0.582·18-s + 0.429·19-s − 3.63·20-s + 0.259·21-s − 2.99·22-s − 1.31·23-s − 1.05·24-s + 2.13·25-s − 3.22·26-s + 0.192·27-s + 0.921·28-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 \( 1 + 158.T + 8.19e3T^{2} \)
5 \( 1 + 6.19e4T + 1.22e9T^{2} \)
7 \( 1 - 1.40e5T + 9.68e10T^{2} \)
11 \( 1 - 1.00e7T + 3.45e13T^{2} \)
13 \( 1 - 3.21e7T + 3.02e14T^{2} \)
17 \( 1 + 1.32e8T + 9.90e15T^{2} \)
19 \( 1 - 8.81e7T + 4.20e16T^{2} \)
23 \( 1 + 9.30e8T + 5.04e17T^{2} \)
29 \( 1 + 2.53e9T + 1.02e19T^{2} \)
31 \( 1 + 1.90e9T + 2.44e19T^{2} \)
37 \( 1 + 5.98e9T + 2.43e20T^{2} \)
41 \( 1 + 3.51e10T + 9.25e20T^{2} \)
43 \( 1 + 5.08e9T + 1.71e21T^{2} \)
47 \( 1 - 5.23e10T + 5.46e21T^{2} \)
53 \( 1 - 6.77e10T + 2.60e22T^{2} \)
61 \( 1 + 3.97e11T + 1.61e23T^{2} \)
67 \( 1 - 1.31e12T + 5.48e23T^{2} \)
71 \( 1 - 2.06e12T + 1.16e24T^{2} \)
73 \( 1 + 3.72e11T + 1.67e24T^{2} \)
79 \( 1 + 2.77e12T + 4.66e24T^{2} \)
83 \( 1 - 9.61e11T + 8.87e24T^{2} \)
89 \( 1 + 1.40e12T + 2.19e25T^{2} \)
97 \( 1 + 1.59e12T + 6.73e25T^{2} \)
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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

−9.468549137088222538999531111855, −8.530001876082736762252924379076, −8.341459752821103581082046949788, −7.22283114022188341372463083774, −6.46591437470645855845858598418, −4.12445226870172347382737872084, −3.54342360112812381628351886025, −1.83927623951646823494365817283, −1.05121683361578666894067991408, 0, 1.05121683361578666894067991408, 1.83927623951646823494365817283, 3.54342360112812381628351886025, 4.12445226870172347382737872084, 6.46591437470645855845858598418, 7.22283114022188341372463083774, 8.341459752821103581082046949788, 8.530001876082736762252924379076, 9.468549137088222538999531111855

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