Properties

Label 2-177-1.1-c13-0-50
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  + 28.6·2-s + 729·3-s − 7.37e3·4-s − 9.30e3·5-s + 2.08e4·6-s + 2.16e5·7-s − 4.46e5·8-s + 5.31e5·9-s − 2.66e5·10-s + 5.39e6·11-s − 5.37e6·12-s + 2.36e7·13-s + 6.20e6·14-s − 6.78e6·15-s + 4.75e7·16-s + 9.98e7·17-s + 1.52e7·18-s + 2.92e8·19-s + 6.86e7·20-s + 1.57e8·21-s + 1.54e8·22-s − 6.28e8·23-s − 3.25e8·24-s − 1.13e9·25-s + 6.79e8·26-s + 3.87e8·27-s − 1.59e9·28-s + ⋯
L(s)  = 1  + 0.316·2-s + 0.577·3-s − 0.899·4-s − 0.266·5-s + 0.182·6-s + 0.695·7-s − 0.601·8-s + 0.333·9-s − 0.0843·10-s + 0.918·11-s − 0.519·12-s + 1.36·13-s + 0.220·14-s − 0.153·15-s + 0.709·16-s + 1.00·17-s + 0.105·18-s + 1.42·19-s + 0.239·20-s + 0.401·21-s + 0.291·22-s − 0.884·23-s − 0.347·24-s − 0.928·25-s + 0.431·26-s + 0.192·27-s − 0.625·28-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

\(L(7)\) \(\approx\) \(3.492872063\)
\(L(\frac12)\) \(\approx\) \(3.492872063\)
\(L(\frac{15}{2})\) 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 \( 1 - 28.6T + 8.19e3T^{2} \)
5 \( 1 + 9.30e3T + 1.22e9T^{2} \)
7 \( 1 - 2.16e5T + 9.68e10T^{2} \)
11 \( 1 - 5.39e6T + 3.45e13T^{2} \)
13 \( 1 - 2.36e7T + 3.02e14T^{2} \)
17 \( 1 - 9.98e7T + 9.90e15T^{2} \)
19 \( 1 - 2.92e8T + 4.20e16T^{2} \)
23 \( 1 + 6.28e8T + 5.04e17T^{2} \)
29 \( 1 + 1.36e9T + 1.02e19T^{2} \)
31 \( 1 - 3.47e8T + 2.44e19T^{2} \)
37 \( 1 + 2.74e9T + 2.43e20T^{2} \)
41 \( 1 + 3.57e10T + 9.25e20T^{2} \)
43 \( 1 + 4.48e10T + 1.71e21T^{2} \)
47 \( 1 - 1.39e11T + 5.46e21T^{2} \)
53 \( 1 + 5.29e9T + 2.60e22T^{2} \)
61 \( 1 - 3.45e11T + 1.61e23T^{2} \)
67 \( 1 + 5.99e11T + 5.48e23T^{2} \)
71 \( 1 + 6.16e11T + 1.16e24T^{2} \)
73 \( 1 - 1.52e12T + 1.67e24T^{2} \)
79 \( 1 - 2.64e12T + 4.66e24T^{2} \)
83 \( 1 - 3.15e12T + 8.87e24T^{2} \)
89 \( 1 - 5.43e12T + 2.19e25T^{2} \)
97 \( 1 + 2.80e12T + 6.73e25T^{2} \)
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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

−10.13000029121582776070123567130, −9.231977597591575057697068197112, −8.371821056088060397120347058803, −7.62437963611071832717149608771, −6.08537577368903751776773102299, −5.08484398190044031312900029237, −3.83736899365247573146194789238, −3.45938503991247173468472917444, −1.65837915097818559373005534269, −0.802295579230023349482423571110, 0.802295579230023349482423571110, 1.65837915097818559373005534269, 3.45938503991247173468472917444, 3.83736899365247573146194789238, 5.08484398190044031312900029237, 6.08537577368903751776773102299, 7.62437963611071832717149608771, 8.371821056088060397120347058803, 9.231977597591575057697068197112, 10.13000029121582776070123567130

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