Properties

Label 2-177-1.1-c13-0-56
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  + 127.·2-s + 729·3-s + 7.97e3·4-s − 1.58e4·5-s + 9.26e4·6-s + 3.38e5·7-s − 2.77e4·8-s + 5.31e5·9-s − 2.00e6·10-s − 9.34e6·11-s + 5.81e6·12-s + 1.20e6·13-s + 4.30e7·14-s − 1.15e7·15-s − 6.88e7·16-s + 8.21e7·17-s + 6.75e7·18-s − 8.01e6·19-s − 1.26e8·20-s + 2.46e8·21-s − 1.18e9·22-s + 1.37e9·23-s − 2.02e7·24-s − 9.71e8·25-s + 1.52e8·26-s + 3.87e8·27-s + 2.69e9·28-s + ⋯
L(s)  = 1  + 1.40·2-s + 0.577·3-s + 0.973·4-s − 0.452·5-s + 0.811·6-s + 1.08·7-s − 0.0373·8-s + 0.333·9-s − 0.635·10-s − 1.59·11-s + 0.561·12-s + 0.0690·13-s + 1.52·14-s − 0.261·15-s − 1.02·16-s + 0.825·17-s + 0.468·18-s − 0.0390·19-s − 0.440·20-s + 0.627·21-s − 2.23·22-s + 1.93·23-s − 0.0215·24-s − 0.795·25-s + 0.0969·26-s + 0.192·27-s + 1.05·28-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

\(L(7)\) \(\approx\) \(6.329817082\)
\(L(\frac12)\) \(\approx\) \(6.329817082\)
\(L(\frac{15}{2})\) 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 \( 1 - 127.T + 8.19e3T^{2} \)
5 \( 1 + 1.58e4T + 1.22e9T^{2} \)
7 \( 1 - 3.38e5T + 9.68e10T^{2} \)
11 \( 1 + 9.34e6T + 3.45e13T^{2} \)
13 \( 1 - 1.20e6T + 3.02e14T^{2} \)
17 \( 1 - 8.21e7T + 9.90e15T^{2} \)
19 \( 1 + 8.01e6T + 4.20e16T^{2} \)
23 \( 1 - 1.37e9T + 5.04e17T^{2} \)
29 \( 1 + 3.06e9T + 1.02e19T^{2} \)
31 \( 1 - 7.91e9T + 2.44e19T^{2} \)
37 \( 1 - 2.32e10T + 2.43e20T^{2} \)
41 \( 1 - 5.29e10T + 9.25e20T^{2} \)
43 \( 1 - 3.43e10T + 1.71e21T^{2} \)
47 \( 1 - 4.30e10T + 5.46e21T^{2} \)
53 \( 1 - 8.06e10T + 2.60e22T^{2} \)
61 \( 1 + 2.15e11T + 1.61e23T^{2} \)
67 \( 1 + 8.36e11T + 5.48e23T^{2} \)
71 \( 1 + 6.69e11T + 1.16e24T^{2} \)
73 \( 1 - 1.21e12T + 1.67e24T^{2} \)
79 \( 1 - 3.05e12T + 4.66e24T^{2} \)
83 \( 1 - 1.99e12T + 8.87e24T^{2} \)
89 \( 1 + 8.83e11T + 2.19e25T^{2} \)
97 \( 1 + 1.28e12T + 6.73e25T^{2} \)
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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

−10.71300171049261639941935517895, −9.267696773967103742092413713746, −8.003975074025205748245634366782, −7.46502901290492405648482714900, −5.86286406426699814918238698680, −4.97359532604060104239521081810, −4.24920861206519987653812314797, −3.05898247020770636106068047990, −2.35592171536863213030889524891, −0.836893784627282030117701837824, 0.836893784627282030117701837824, 2.35592171536863213030889524891, 3.05898247020770636106068047990, 4.24920861206519987653812314797, 4.97359532604060104239521081810, 5.86286406426699814918238698680, 7.46502901290492405648482714900, 8.003975074025205748245634366782, 9.267696773967103742092413713746, 10.71300171049261639941935517895

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