Properties

Label 2-177-1.1-c13-0-88
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  + 117.·2-s + 729·3-s + 5.69e3·4-s + 4.76e4·5-s + 8.59e4·6-s − 6.74e4·7-s − 2.94e5·8-s + 5.31e5·9-s + 5.62e6·10-s + 9.45e6·11-s + 4.15e6·12-s + 2.98e7·13-s − 7.94e6·14-s + 3.47e7·15-s − 8.13e7·16-s + 1.61e8·17-s + 6.26e7·18-s − 3.34e8·19-s + 2.71e8·20-s − 4.91e7·21-s + 1.11e9·22-s + 9.56e8·23-s − 2.14e8·24-s + 1.05e9·25-s + 3.51e9·26-s + 3.87e8·27-s − 3.84e8·28-s + ⋯
L(s)  = 1  + 1.30·2-s + 0.577·3-s + 0.695·4-s + 1.36·5-s + 0.751·6-s − 0.216·7-s − 0.396·8-s + 0.333·9-s + 1.77·10-s + 1.60·11-s + 0.401·12-s + 1.71·13-s − 0.281·14-s + 0.788·15-s − 1.21·16-s + 1.62·17-s + 0.434·18-s − 1.63·19-s + 0.949·20-s − 0.125·21-s + 2.09·22-s + 1.34·23-s − 0.228·24-s + 0.863·25-s + 2.23·26-s + 0.192·27-s − 0.150·28-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

\(L(7)\) \(\approx\) \(9.378350528\)
\(L(\frac12)\) \(\approx\) \(9.378350528\)
\(L(\frac{15}{2})\) 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 \( 1 - 117.T + 8.19e3T^{2} \)
5 \( 1 - 4.76e4T + 1.22e9T^{2} \)
7 \( 1 + 6.74e4T + 9.68e10T^{2} \)
11 \( 1 - 9.45e6T + 3.45e13T^{2} \)
13 \( 1 - 2.98e7T + 3.02e14T^{2} \)
17 \( 1 - 1.61e8T + 9.90e15T^{2} \)
19 \( 1 + 3.34e8T + 4.20e16T^{2} \)
23 \( 1 - 9.56e8T + 5.04e17T^{2} \)
29 \( 1 + 3.57e9T + 1.02e19T^{2} \)
31 \( 1 - 2.34e9T + 2.44e19T^{2} \)
37 \( 1 + 3.01e10T + 2.43e20T^{2} \)
41 \( 1 + 3.36e10T + 9.25e20T^{2} \)
43 \( 1 - 6.11e10T + 1.71e21T^{2} \)
47 \( 1 + 7.41e10T + 5.46e21T^{2} \)
53 \( 1 - 2.55e11T + 2.60e22T^{2} \)
61 \( 1 - 2.16e11T + 1.61e23T^{2} \)
67 \( 1 - 1.78e11T + 5.48e23T^{2} \)
71 \( 1 - 1.58e12T + 1.16e24T^{2} \)
73 \( 1 - 7.01e11T + 1.67e24T^{2} \)
79 \( 1 + 3.81e12T + 4.66e24T^{2} \)
83 \( 1 - 2.60e11T + 8.87e24T^{2} \)
89 \( 1 - 6.27e12T + 2.19e25T^{2} \)
97 \( 1 - 6.89e12T + 6.73e25T^{2} \)
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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

−10.34287454753168603265840310607, −9.203285560670453139587189379486, −8.670637535704938732314971492983, −6.72299640265625330804990182467, −6.16277029998667020717966958303, −5.25496929366587610630876736991, −3.86353297544823866237722896216, −3.34845237783998218341721956580, −1.98760908074501929434795370570, −1.15028347557634815043494666376, 1.15028347557634815043494666376, 1.98760908074501929434795370570, 3.34845237783998218341721956580, 3.86353297544823866237722896216, 5.25496929366587610630876736991, 6.16277029998667020717966958303, 6.72299640265625330804990182467, 8.670637535704938732314971492983, 9.203285560670453139587189379486, 10.34287454753168603265840310607

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