Properties

Label 2-177-1.1-c13-0-77
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  + 84.6·2-s + 729·3-s − 1.02e3·4-s + 3.30e4·5-s + 6.17e4·6-s + 5.92e5·7-s − 7.80e5·8-s + 5.31e5·9-s + 2.80e6·10-s + 7.93e6·11-s − 7.44e5·12-s − 3.12e7·13-s + 5.02e7·14-s + 2.41e7·15-s − 5.76e7·16-s + 7.78e7·17-s + 4.50e7·18-s + 4.96e7·19-s − 3.37e7·20-s + 4.32e8·21-s + 6.71e8·22-s + 3.26e8·23-s − 5.68e8·24-s − 1.26e8·25-s − 2.64e9·26-s + 3.87e8·27-s − 6.05e8·28-s + ⋯
L(s)  = 1  + 0.935·2-s + 0.577·3-s − 0.124·4-s + 0.946·5-s + 0.540·6-s + 1.90·7-s − 1.05·8-s + 0.333·9-s + 0.885·10-s + 1.35·11-s − 0.0719·12-s − 1.79·13-s + 1.78·14-s + 0.546·15-s − 0.859·16-s + 0.782·17-s + 0.311·18-s + 0.242·19-s − 0.118·20-s + 1.09·21-s + 1.26·22-s + 0.460·23-s − 0.607·24-s − 0.103·25-s − 1.67·26-s + 0.192·27-s − 0.237·28-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

\(L(7)\) \(\approx\) \(7.181335725\)
\(L(\frac12)\) \(\approx\) \(7.181335725\)
\(L(\frac{15}{2})\) 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 \( 1 - 84.6T + 8.19e3T^{2} \)
5 \( 1 - 3.30e4T + 1.22e9T^{2} \)
7 \( 1 - 5.92e5T + 9.68e10T^{2} \)
11 \( 1 - 7.93e6T + 3.45e13T^{2} \)
13 \( 1 + 3.12e7T + 3.02e14T^{2} \)
17 \( 1 - 7.78e7T + 9.90e15T^{2} \)
19 \( 1 - 4.96e7T + 4.20e16T^{2} \)
23 \( 1 - 3.26e8T + 5.04e17T^{2} \)
29 \( 1 - 3.75e9T + 1.02e19T^{2} \)
31 \( 1 - 5.58e9T + 2.44e19T^{2} \)
37 \( 1 - 1.24e10T + 2.43e20T^{2} \)
41 \( 1 + 5.31e10T + 9.25e20T^{2} \)
43 \( 1 - 1.81e10T + 1.71e21T^{2} \)
47 \( 1 - 4.47e10T + 5.46e21T^{2} \)
53 \( 1 + 6.00e10T + 2.60e22T^{2} \)
61 \( 1 + 4.44e11T + 1.61e23T^{2} \)
67 \( 1 + 6.55e11T + 5.48e23T^{2} \)
71 \( 1 + 8.05e11T + 1.16e24T^{2} \)
73 \( 1 + 1.53e12T + 1.67e24T^{2} \)
79 \( 1 - 1.06e12T + 4.66e24T^{2} \)
83 \( 1 + 3.39e12T + 8.87e24T^{2} \)
89 \( 1 - 7.72e12T + 2.19e25T^{2} \)
97 \( 1 + 1.94e12T + 6.73e25T^{2} \)
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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

−10.18064605639500976770389354607, −9.334815193990277420990159367976, −8.412757985206218839651183898115, −7.30052978728722657429759930010, −5.96723594367675350624859532145, −4.87310500372794719459897433969, −4.46403775043207507645112017255, −2.98801696846975694940135594511, −1.97396854348878150722748863903, −1.03838023125052539767212916470, 1.03838023125052539767212916470, 1.97396854348878150722748863903, 2.98801696846975694940135594511, 4.46403775043207507645112017255, 4.87310500372794719459897433969, 5.96723594367675350624859532145, 7.30052978728722657429759930010, 8.412757985206218839651183898115, 9.334815193990277420990159367976, 10.18064605639500976770389354607

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