Properties

Label 2-177-1.1-c13-0-4
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 107.·2-s + 729·3-s + 3.46e3·4-s + 5.61e3·5-s − 7.87e4·6-s − 4.41e5·7-s + 5.09e5·8-s + 5.31e5·9-s − 6.06e5·10-s − 9.58e6·11-s + 2.52e6·12-s − 1.04e6·13-s + 4.76e7·14-s + 4.09e6·15-s − 8.34e7·16-s + 8.94e7·17-s − 5.73e7·18-s − 1.31e8·19-s + 1.94e7·20-s − 3.21e8·21-s + 1.03e9·22-s − 6.51e8·23-s + 3.71e8·24-s − 1.18e9·25-s + 1.12e8·26-s + 3.87e8·27-s − 1.53e9·28-s + ⋯
L(s)  = 1  − 1.19·2-s + 0.577·3-s + 0.423·4-s + 0.160·5-s − 0.688·6-s − 1.41·7-s + 0.687·8-s + 0.333·9-s − 0.191·10-s − 1.63·11-s + 0.244·12-s − 0.0598·13-s + 1.69·14-s + 0.0927·15-s − 1.24·16-s + 0.898·17-s − 0.397·18-s − 0.642·19-s + 0.0680·20-s − 0.818·21-s + 1.94·22-s − 0.917·23-s + 0.397·24-s − 0.974·25-s + 0.0713·26-s + 0.192·27-s − 0.600·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(0.1397302836\)
\(L(\frac12)\) \(\approx\) \(0.1397302836\)
\(L(\frac{15}{2})\) 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 \( 1 + 107.T + 8.19e3T^{2} \)
5 \( 1 - 5.61e3T + 1.22e9T^{2} \)
7 \( 1 + 4.41e5T + 9.68e10T^{2} \)
11 \( 1 + 9.58e6T + 3.45e13T^{2} \)
13 \( 1 + 1.04e6T + 3.02e14T^{2} \)
17 \( 1 - 8.94e7T + 9.90e15T^{2} \)
19 \( 1 + 1.31e8T + 4.20e16T^{2} \)
23 \( 1 + 6.51e8T + 5.04e17T^{2} \)
29 \( 1 + 4.82e9T + 1.02e19T^{2} \)
31 \( 1 - 6.29e9T + 2.44e19T^{2} \)
37 \( 1 + 1.75e10T + 2.43e20T^{2} \)
41 \( 1 + 3.34e10T + 9.25e20T^{2} \)
43 \( 1 - 1.01e10T + 1.71e21T^{2} \)
47 \( 1 + 7.21e10T + 5.46e21T^{2} \)
53 \( 1 + 1.09e11T + 2.60e22T^{2} \)
61 \( 1 + 2.31e11T + 1.61e23T^{2} \)
67 \( 1 - 1.73e11T + 5.48e23T^{2} \)
71 \( 1 - 1.31e12T + 1.16e24T^{2} \)
73 \( 1 - 1.72e12T + 1.67e24T^{2} \)
79 \( 1 + 2.25e12T + 4.66e24T^{2} \)
83 \( 1 - 2.25e12T + 8.87e24T^{2} \)
89 \( 1 + 3.90e12T + 2.19e25T^{2} \)
97 \( 1 + 9.75e12T + 6.73e25T^{2} \)
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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

−9.908844586171569599448654761397, −9.605226420234069417888168417257, −8.307084570165148012039558296665, −7.75358516943864713659708161960, −6.61739113995526397062241857294, −5.33717455392007927838078871377, −3.81442638248276381287122386478, −2.72020780821669169749221874574, −1.71058762880153823804227731767, −0.17756855605938558619263484376, 0.17756855605938558619263484376, 1.71058762880153823804227731767, 2.72020780821669169749221874574, 3.81442638248276381287122386478, 5.33717455392007927838078871377, 6.61739113995526397062241857294, 7.75358516943864713659708161960, 8.307084570165148012039558296665, 9.605226420234069417888168417257, 9.908844586171569599448654761397

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