Properties

Label 2-177-1.1-c13-0-27
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  − 7.03·2-s + 729·3-s − 8.14e3·4-s + 1.24e4·5-s − 5.12e3·6-s − 1.79e5·7-s + 1.14e5·8-s + 5.31e5·9-s − 8.77e4·10-s − 8.91e6·11-s − 5.93e6·12-s + 2.94e7·13-s + 1.26e6·14-s + 9.08e6·15-s + 6.58e7·16-s − 1.63e7·17-s − 3.73e6·18-s − 1.07e8·19-s − 1.01e8·20-s − 1.30e8·21-s + 6.26e7·22-s + 5.82e8·23-s + 8.37e7·24-s − 1.06e9·25-s − 2.06e8·26-s + 3.87e8·27-s + 1.46e9·28-s + ⋯
L(s)  = 1  − 0.0777·2-s + 0.577·3-s − 0.993·4-s + 0.356·5-s − 0.0448·6-s − 0.577·7-s + 0.154·8-s + 0.333·9-s − 0.0277·10-s − 1.51·11-s − 0.573·12-s + 1.69·13-s + 0.0448·14-s + 0.205·15-s + 0.981·16-s − 0.164·17-s − 0.0259·18-s − 0.522·19-s − 0.354·20-s − 0.333·21-s + 0.117·22-s + 0.819·23-s + 0.0894·24-s − 0.872·25-s − 0.131·26-s + 0.192·27-s + 0.573·28-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

\(L(7)\) \(\approx\) \(1.498788871\)
\(L(\frac12)\) \(\approx\) \(1.498788871\)
\(L(\frac{15}{2})\) 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 \( 1 + 7.03T + 8.19e3T^{2} \)
5 \( 1 - 1.24e4T + 1.22e9T^{2} \)
7 \( 1 + 1.79e5T + 9.68e10T^{2} \)
11 \( 1 + 8.91e6T + 3.45e13T^{2} \)
13 \( 1 - 2.94e7T + 3.02e14T^{2} \)
17 \( 1 + 1.63e7T + 9.90e15T^{2} \)
19 \( 1 + 1.07e8T + 4.20e16T^{2} \)
23 \( 1 - 5.82e8T + 5.04e17T^{2} \)
29 \( 1 + 1.37e9T + 1.02e19T^{2} \)
31 \( 1 + 5.70e9T + 2.44e19T^{2} \)
37 \( 1 - 6.03e9T + 2.43e20T^{2} \)
41 \( 1 + 1.15e10T + 9.25e20T^{2} \)
43 \( 1 + 1.85e9T + 1.71e21T^{2} \)
47 \( 1 - 4.15e9T + 5.46e21T^{2} \)
53 \( 1 - 1.45e11T + 2.60e22T^{2} \)
61 \( 1 + 4.25e11T + 1.61e23T^{2} \)
67 \( 1 - 2.63e11T + 5.48e23T^{2} \)
71 \( 1 - 3.12e11T + 1.16e24T^{2} \)
73 \( 1 + 7.30e11T + 1.67e24T^{2} \)
79 \( 1 - 2.67e12T + 4.66e24T^{2} \)
83 \( 1 + 1.18e12T + 8.87e24T^{2} \)
89 \( 1 - 3.66e11T + 2.19e25T^{2} \)
97 \( 1 - 1.46e13T + 6.73e25T^{2} \)
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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

−10.18980910989254604256860839135, −9.229251741845022599842357232386, −8.510793106974239514890524979902, −7.60683687189234236288385583995, −6.14578654204890157967901389422, −5.20446609081908462295652045125, −3.94998389207637356061904977221, −3.10180466722087979607120288104, −1.79596882339288077597960801087, −0.50899090569656777369038356039, 0.50899090569656777369038356039, 1.79596882339288077597960801087, 3.10180466722087979607120288104, 3.94998389207637356061904977221, 5.20446609081908462295652045125, 6.14578654204890157967901389422, 7.60683687189234236288385583995, 8.510793106974239514890524979902, 9.229251741845022599842357232386, 10.18980910989254604256860839135

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