Properties

Label 2-177-1.1-c13-0-48
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  + 66.7·2-s + 729·3-s − 3.73e3·4-s + 3.60e4·5-s + 4.86e4·6-s − 2.69e5·7-s − 7.96e5·8-s + 5.31e5·9-s + 2.40e6·10-s + 1.04e7·11-s − 2.71e6·12-s − 7.55e6·13-s − 1.80e7·14-s + 2.62e7·15-s − 2.26e7·16-s − 1.33e8·17-s + 3.54e7·18-s + 2.81e8·19-s − 1.34e8·20-s − 1.96e8·21-s + 6.94e8·22-s + 2.81e8·23-s − 5.80e8·24-s + 8.07e7·25-s − 5.04e8·26-s + 3.87e8·27-s + 1.00e9·28-s + ⋯
L(s)  = 1  + 0.738·2-s + 0.577·3-s − 0.455·4-s + 1.03·5-s + 0.426·6-s − 0.866·7-s − 1.07·8-s + 0.333·9-s + 0.762·10-s + 1.77·11-s − 0.262·12-s − 0.434·13-s − 0.639·14-s + 0.596·15-s − 0.337·16-s − 1.33·17-s + 0.246·18-s + 1.37·19-s − 0.470·20-s − 0.500·21-s + 1.30·22-s + 0.397·23-s − 0.620·24-s + 0.0661·25-s − 0.320·26-s + 0.192·27-s + 0.394·28-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

\(L(7)\) \(\approx\) \(4.173153114\)
\(L(\frac12)\) \(\approx\) \(4.173153114\)
\(L(\frac{15}{2})\) 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 \( 1 - 66.7T + 8.19e3T^{2} \)
5 \( 1 - 3.60e4T + 1.22e9T^{2} \)
7 \( 1 + 2.69e5T + 9.68e10T^{2} \)
11 \( 1 - 1.04e7T + 3.45e13T^{2} \)
13 \( 1 + 7.55e6T + 3.02e14T^{2} \)
17 \( 1 + 1.33e8T + 9.90e15T^{2} \)
19 \( 1 - 2.81e8T + 4.20e16T^{2} \)
23 \( 1 - 2.81e8T + 5.04e17T^{2} \)
29 \( 1 - 3.19e9T + 1.02e19T^{2} \)
31 \( 1 + 5.46e9T + 2.44e19T^{2} \)
37 \( 1 + 7.12e9T + 2.43e20T^{2} \)
41 \( 1 - 2.84e10T + 9.25e20T^{2} \)
43 \( 1 + 4.95e10T + 1.71e21T^{2} \)
47 \( 1 + 1.09e10T + 5.46e21T^{2} \)
53 \( 1 - 2.87e11T + 2.60e22T^{2} \)
61 \( 1 - 1.11e11T + 1.61e23T^{2} \)
67 \( 1 - 9.50e11T + 5.48e23T^{2} \)
71 \( 1 - 2.78e11T + 1.16e24T^{2} \)
73 \( 1 - 5.64e11T + 1.67e24T^{2} \)
79 \( 1 - 1.36e12T + 4.66e24T^{2} \)
83 \( 1 - 2.62e12T + 8.87e24T^{2} \)
89 \( 1 + 7.86e12T + 2.19e25T^{2} \)
97 \( 1 - 8.83e12T + 6.73e25T^{2} \)
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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

−9.912217947115501011382209601567, −9.355336376987430978168375880173, −8.769409232648173950624326153065, −6.94714291183350628070367887970, −6.24449197528836492981130606512, −5.13233111938152133369004841920, −3.99333138043340226405600970395, −3.17301361576883420917188101207, −2.03202068427665547557624912273, −0.75699411441656317557277869526, 0.75699411441656317557277869526, 2.03202068427665547557624912273, 3.17301361576883420917188101207, 3.99333138043340226405600970395, 5.13233111938152133369004841920, 6.24449197528836492981130606512, 6.94714291183350628070367887970, 8.769409232648173950624326153065, 9.355336376987430978168375880173, 9.912217947115501011382209601567

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