Properties

Label 2-177-1.1-c13-0-38
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  + 118.·2-s + 729·3-s + 5.83e3·4-s − 5.41e4·5-s + 8.63e4·6-s + 1.34e5·7-s − 2.78e5·8-s + 5.31e5·9-s − 6.41e6·10-s + 2.87e6·11-s + 4.25e6·12-s + 1.21e7·13-s + 1.59e7·14-s − 3.94e7·15-s − 8.08e7·16-s − 9.77e7·17-s + 6.29e7·18-s − 2.15e8·19-s − 3.16e8·20-s + 9.81e7·21-s + 3.40e8·22-s + 9.00e7·23-s − 2.03e8·24-s + 1.71e9·25-s + 1.43e9·26-s + 3.87e8·27-s + 7.85e8·28-s + ⋯
L(s)  = 1  + 1.30·2-s + 0.577·3-s + 0.712·4-s − 1.55·5-s + 0.755·6-s + 0.432·7-s − 0.376·8-s + 0.333·9-s − 2.02·10-s + 0.489·11-s + 0.411·12-s + 0.696·13-s + 0.566·14-s − 0.895·15-s − 1.20·16-s − 0.982·17-s + 0.436·18-s − 1.04·19-s − 1.10·20-s + 0.249·21-s + 0.640·22-s + 0.126·23-s − 0.217·24-s + 1.40·25-s + 0.911·26-s + 0.192·27-s + 0.308·28-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

\(L(7)\) \(\approx\) \(3.820582980\)
\(L(\frac12)\) \(\approx\) \(3.820582980\)
\(L(\frac{15}{2})\) 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 \( 1 - 118.T + 8.19e3T^{2} \)
5 \( 1 + 5.41e4T + 1.22e9T^{2} \)
7 \( 1 - 1.34e5T + 9.68e10T^{2} \)
11 \( 1 - 2.87e6T + 3.45e13T^{2} \)
13 \( 1 - 1.21e7T + 3.02e14T^{2} \)
17 \( 1 + 9.77e7T + 9.90e15T^{2} \)
19 \( 1 + 2.15e8T + 4.20e16T^{2} \)
23 \( 1 - 9.00e7T + 5.04e17T^{2} \)
29 \( 1 - 5.19e9T + 1.02e19T^{2} \)
31 \( 1 + 6.40e9T + 2.44e19T^{2} \)
37 \( 1 - 8.47e9T + 2.43e20T^{2} \)
41 \( 1 + 6.71e8T + 9.25e20T^{2} \)
43 \( 1 - 4.97e10T + 1.71e21T^{2} \)
47 \( 1 - 8.08e10T + 5.46e21T^{2} \)
53 \( 1 - 2.96e10T + 2.60e22T^{2} \)
61 \( 1 - 3.60e11T + 1.61e23T^{2} \)
67 \( 1 - 8.86e11T + 5.48e23T^{2} \)
71 \( 1 + 1.90e12T + 1.16e24T^{2} \)
73 \( 1 - 9.74e11T + 1.67e24T^{2} \)
79 \( 1 - 1.54e11T + 4.66e24T^{2} \)
83 \( 1 + 1.19e11T + 8.87e24T^{2} \)
89 \( 1 - 7.57e12T + 2.19e25T^{2} \)
97 \( 1 - 4.06e12T + 6.73e25T^{2} \)
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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

−10.82705047490124603122741532368, −8.974370759888770814297665869000, −8.365944727334009023033526426833, −7.19786963879808686246379174921, −6.21124160394953513891062617713, −4.67978041763853051411230588590, −4.13022971944774755061899995747, −3.40699429085513728035573233151, −2.23961645943676231820595839935, −0.64484608271404752827903895320, 0.64484608271404752827903895320, 2.23961645943676231820595839935, 3.40699429085513728035573233151, 4.13022971944774755061899995747, 4.67978041763853051411230588590, 6.21124160394953513891062617713, 7.19786963879808686246379174921, 8.365944727334009023033526426833, 8.974370759888770814297665869000, 10.82705047490124603122741532368

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