Properties

Label 2-177-1.1-c13-0-123
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 166.·2-s + 729·3-s + 1.94e4·4-s + 2.16e4·5-s + 1.21e5·6-s − 5.57e5·7-s + 1.87e6·8-s + 5.31e5·9-s + 3.60e6·10-s − 6.15e6·11-s + 1.41e7·12-s + 7.79e6·13-s − 9.27e7·14-s + 1.58e7·15-s + 1.52e8·16-s − 7.99e7·17-s + 8.83e7·18-s − 3.34e8·19-s + 4.21e8·20-s − 4.06e8·21-s − 1.02e9·22-s − 2.66e8·23-s + 1.36e9·24-s − 7.50e8·25-s + 1.29e9·26-s + 3.87e8·27-s − 1.08e10·28-s + ⋯
L(s)  = 1  + 1.83·2-s + 0.577·3-s + 2.37·4-s + 0.620·5-s + 1.06·6-s − 1.79·7-s + 2.52·8-s + 0.333·9-s + 1.13·10-s − 1.04·11-s + 1.37·12-s + 0.447·13-s − 3.29·14-s + 0.358·15-s + 2.26·16-s − 0.803·17-s + 0.612·18-s − 1.63·19-s + 1.47·20-s − 1.03·21-s − 1.92·22-s − 0.375·23-s + 1.45·24-s − 0.615·25-s + 0.822·26-s + 0.192·27-s − 4.25·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) 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 \( 1 - 166.T + 8.19e3T^{2} \)
5 \( 1 - 2.16e4T + 1.22e9T^{2} \)
7 \( 1 + 5.57e5T + 9.68e10T^{2} \)
11 \( 1 + 6.15e6T + 3.45e13T^{2} \)
13 \( 1 - 7.79e6T + 3.02e14T^{2} \)
17 \( 1 + 7.99e7T + 9.90e15T^{2} \)
19 \( 1 + 3.34e8T + 4.20e16T^{2} \)
23 \( 1 + 2.66e8T + 5.04e17T^{2} \)
29 \( 1 - 1.98e9T + 1.02e19T^{2} \)
31 \( 1 + 4.30e9T + 2.44e19T^{2} \)
37 \( 1 + 1.90e10T + 2.43e20T^{2} \)
41 \( 1 - 5.67e10T + 9.25e20T^{2} \)
43 \( 1 - 5.42e10T + 1.71e21T^{2} \)
47 \( 1 + 1.35e11T + 5.46e21T^{2} \)
53 \( 1 - 1.81e11T + 2.60e22T^{2} \)
61 \( 1 + 6.00e11T + 1.61e23T^{2} \)
67 \( 1 + 1.07e12T + 5.48e23T^{2} \)
71 \( 1 + 1.23e12T + 1.16e24T^{2} \)
73 \( 1 - 1.67e12T + 1.67e24T^{2} \)
79 \( 1 - 7.00e11T + 4.66e24T^{2} \)
83 \( 1 + 1.64e12T + 8.87e24T^{2} \)
89 \( 1 + 5.34e12T + 2.19e25T^{2} \)
97 \( 1 - 2.11e12T + 6.73e25T^{2} \)
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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

−10.13954421972804720431957849799, −8.901517801120471769937588488332, −7.34872060655535319795142333057, −6.31781973001408520342130561336, −5.87298446608070886299602940618, −4.46827983804087168650198551051, −3.56809044493960622632082388663, −2.67170404063197873448477791707, −2.02691168705688749756592615023, 0, 2.02691168705688749756592615023, 2.67170404063197873448477791707, 3.56809044493960622632082388663, 4.46827983804087168650198551051, 5.87298446608070886299602940618, 6.31781973001408520342130561336, 7.34872060655535319795142333057, 8.901517801120471769937588488332, 10.13954421972804720431957849799

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