Properties

Label 2-177-1.1-c13-0-11
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  + 15.7·2-s + 729·3-s − 7.94e3·4-s − 1.82e4·5-s + 1.14e4·6-s − 1.39e5·7-s − 2.54e5·8-s + 5.31e5·9-s − 2.87e5·10-s + 3.23e6·11-s − 5.79e6·12-s − 2.38e7·13-s − 2.19e6·14-s − 1.32e7·15-s + 6.10e7·16-s − 1.01e8·17-s + 8.37e6·18-s − 2.58e8·19-s + 1.44e8·20-s − 1.01e8·21-s + 5.09e7·22-s + 4.75e8·23-s − 1.85e8·24-s − 8.88e8·25-s − 3.76e8·26-s + 3.87e8·27-s + 1.10e9·28-s + ⋯
L(s)  = 1  + 0.174·2-s + 0.577·3-s − 0.969·4-s − 0.521·5-s + 0.100·6-s − 0.448·7-s − 0.343·8-s + 0.333·9-s − 0.0908·10-s + 0.549·11-s − 0.559·12-s − 1.37·13-s − 0.0780·14-s − 0.301·15-s + 0.909·16-s − 1.01·17-s + 0.0580·18-s − 1.26·19-s + 0.506·20-s − 0.258·21-s + 0.0957·22-s + 0.669·23-s − 0.198·24-s − 0.727·25-s − 0.238·26-s + 0.192·27-s + 0.434·28-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

\(L(7)\) \(\approx\) \(0.5936686115\)
\(L(\frac12)\) \(\approx\) \(0.5936686115\)
\(L(\frac{15}{2})\) 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 \( 1 - 15.7T + 8.19e3T^{2} \)
5 \( 1 + 1.82e4T + 1.22e9T^{2} \)
7 \( 1 + 1.39e5T + 9.68e10T^{2} \)
11 \( 1 - 3.23e6T + 3.45e13T^{2} \)
13 \( 1 + 2.38e7T + 3.02e14T^{2} \)
17 \( 1 + 1.01e8T + 9.90e15T^{2} \)
19 \( 1 + 2.58e8T + 4.20e16T^{2} \)
23 \( 1 - 4.75e8T + 5.04e17T^{2} \)
29 \( 1 + 4.16e9T + 1.02e19T^{2} \)
31 \( 1 - 3.82e9T + 2.44e19T^{2} \)
37 \( 1 + 2.63e9T + 2.43e20T^{2} \)
41 \( 1 + 1.28e10T + 9.25e20T^{2} \)
43 \( 1 + 9.65e9T + 1.71e21T^{2} \)
47 \( 1 + 8.50e9T + 5.46e21T^{2} \)
53 \( 1 + 2.27e11T + 2.60e22T^{2} \)
61 \( 1 + 2.60e11T + 1.61e23T^{2} \)
67 \( 1 - 5.68e11T + 5.48e23T^{2} \)
71 \( 1 + 7.25e11T + 1.16e24T^{2} \)
73 \( 1 - 8.77e11T + 1.67e24T^{2} \)
79 \( 1 + 1.57e12T + 4.66e24T^{2} \)
83 \( 1 + 2.61e12T + 8.87e24T^{2} \)
89 \( 1 + 8.27e11T + 2.19e25T^{2} \)
97 \( 1 - 1.71e12T + 6.73e25T^{2} \)
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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

−10.04044379629846404519557031169, −9.283059132254533836110477468768, −8.489274000649385139109860472454, −7.45385857063942125877735875156, −6.33956708675538026891387128199, −4.84741253377822086659692673406, −4.14095092925999023603112372785, −3.15514706900656458552014565997, −1.91444697253846284949381996826, −0.29721261984394390565800647306, 0.29721261984394390565800647306, 1.91444697253846284949381996826, 3.15514706900656458552014565997, 4.14095092925999023603112372785, 4.84741253377822086659692673406, 6.33956708675538026891387128199, 7.45385857063942125877735875156, 8.489274000649385139109860472454, 9.283059132254533836110477468768, 10.04044379629846404519557031169

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