Properties

Label 2-177-1.1-c13-0-0
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  − 60.0·2-s + 729·3-s − 4.58e3·4-s − 5.08e4·5-s − 4.38e4·6-s − 2.59e5·7-s + 7.67e5·8-s + 5.31e5·9-s + 3.05e6·10-s − 1.61e6·11-s − 3.33e6·12-s + 3.32e6·13-s + 1.55e7·14-s − 3.70e7·15-s − 8.60e6·16-s − 9.95e7·17-s − 3.19e7·18-s + 9.24e7·19-s + 2.32e8·20-s − 1.89e8·21-s + 9.71e7·22-s − 5.67e8·23-s + 5.59e8·24-s + 1.36e9·25-s − 1.99e8·26-s + 3.87e8·27-s + 1.18e9·28-s + ⋯
L(s)  = 1  − 0.663·2-s + 0.577·3-s − 0.559·4-s − 1.45·5-s − 0.383·6-s − 0.833·7-s + 1.03·8-s + 0.333·9-s + 0.966·10-s − 0.275·11-s − 0.322·12-s + 0.191·13-s + 0.553·14-s − 0.840·15-s − 0.128·16-s − 1.00·17-s − 0.221·18-s + 0.450·19-s + 0.813·20-s − 0.481·21-s + 0.182·22-s − 0.799·23-s + 0.597·24-s + 1.11·25-s − 0.126·26-s + 0.192·27-s + 0.465·28-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

\(L(7)\) \(\approx\) \(0.006916290542\)
\(L(\frac12)\) \(\approx\) \(0.006916290542\)
\(L(\frac{15}{2})\) 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 \( 1 + 60.0T + 8.19e3T^{2} \)
5 \( 1 + 5.08e4T + 1.22e9T^{2} \)
7 \( 1 + 2.59e5T + 9.68e10T^{2} \)
11 \( 1 + 1.61e6T + 3.45e13T^{2} \)
13 \( 1 - 3.32e6T + 3.02e14T^{2} \)
17 \( 1 + 9.95e7T + 9.90e15T^{2} \)
19 \( 1 - 9.24e7T + 4.20e16T^{2} \)
23 \( 1 + 5.67e8T + 5.04e17T^{2} \)
29 \( 1 - 3.47e9T + 1.02e19T^{2} \)
31 \( 1 + 8.56e9T + 2.44e19T^{2} \)
37 \( 1 + 1.81e10T + 2.43e20T^{2} \)
41 \( 1 + 1.66e8T + 9.25e20T^{2} \)
43 \( 1 + 5.81e10T + 1.71e21T^{2} \)
47 \( 1 + 8.85e10T + 5.46e21T^{2} \)
53 \( 1 + 1.79e11T + 2.60e22T^{2} \)
61 \( 1 + 3.70e11T + 1.61e23T^{2} \)
67 \( 1 + 3.15e11T + 5.48e23T^{2} \)
71 \( 1 + 1.56e12T + 1.16e24T^{2} \)
73 \( 1 - 1.08e12T + 1.67e24T^{2} \)
79 \( 1 - 1.56e12T + 4.66e24T^{2} \)
83 \( 1 - 2.63e12T + 8.87e24T^{2} \)
89 \( 1 + 6.98e12T + 2.19e25T^{2} \)
97 \( 1 + 1.52e13T + 6.73e25T^{2} \)
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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

−10.13881569065312434335574281095, −9.175543910330056321093541183874, −8.379027039698604868499876934881, −7.67179160592346430998721728975, −6.69464732517902341667237793888, −4.90752196642906419840891557554, −3.91768737552045338457350312858, −3.18130229652812377692392654341, −1.58349061485020489044225692603, −0.03830804981828638774134163136, 0.03830804981828638774134163136, 1.58349061485020489044225692603, 3.18130229652812377692392654341, 3.91768737552045338457350312858, 4.90752196642906419840891557554, 6.69464732517902341667237793888, 7.67179160592346430998721728975, 8.379027039698604868499876934881, 9.175543910330056321093541183874, 10.13881569065312434335574281095

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