Properties

Label 2-177-1.1-c11-0-78
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 32.2·2-s + 243·3-s − 1.01e3·4-s − 5.50e3·5-s + 7.82e3·6-s − 3.67e4·7-s − 9.85e4·8-s + 5.90e4·9-s − 1.77e5·10-s + 7.85e5·11-s − 2.45e5·12-s + 1.64e6·13-s − 1.18e6·14-s − 1.33e6·15-s − 1.10e6·16-s + 1.42e4·17-s + 1.90e6·18-s − 8.87e6·19-s + 5.56e6·20-s − 8.92e6·21-s + 2.52e7·22-s + 2.97e7·23-s − 2.39e7·24-s − 1.84e7·25-s + 5.30e7·26-s + 1.43e7·27-s + 3.71e7·28-s + ⋯
L(s)  = 1  + 0.711·2-s + 0.577·3-s − 0.493·4-s − 0.788·5-s + 0.410·6-s − 0.826·7-s − 1.06·8-s + 0.333·9-s − 0.561·10-s + 1.47·11-s − 0.284·12-s + 1.23·13-s − 0.587·14-s − 0.455·15-s − 0.262·16-s + 0.00243·17-s + 0.237·18-s − 0.822·19-s + 0.389·20-s − 0.476·21-s + 1.04·22-s + 0.962·23-s − 0.613·24-s − 0.378·25-s + 0.876·26-s + 0.192·27-s + 0.407·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 243T \)
59 \( 1 + 7.14e8T \)
good2 \( 1 - 32.2T + 2.04e3T^{2} \)
5 \( 1 + 5.50e3T + 4.88e7T^{2} \)
7 \( 1 + 3.67e4T + 1.97e9T^{2} \)
11 \( 1 - 7.85e5T + 2.85e11T^{2} \)
13 \( 1 - 1.64e6T + 1.79e12T^{2} \)
17 \( 1 - 1.42e4T + 3.42e13T^{2} \)
19 \( 1 + 8.87e6T + 1.16e14T^{2} \)
23 \( 1 - 2.97e7T + 9.52e14T^{2} \)
29 \( 1 - 6.76e7T + 1.22e16T^{2} \)
31 \( 1 - 6.05e7T + 2.54e16T^{2} \)
37 \( 1 - 1.57e8T + 1.77e17T^{2} \)
41 \( 1 + 1.38e9T + 5.50e17T^{2} \)
43 \( 1 + 1.09e9T + 9.29e17T^{2} \)
47 \( 1 - 1.74e9T + 2.47e18T^{2} \)
53 \( 1 - 3.03e8T + 9.26e18T^{2} \)
61 \( 1 - 1.62e9T + 4.35e19T^{2} \)
67 \( 1 + 1.24e10T + 1.22e20T^{2} \)
71 \( 1 + 8.29e9T + 2.31e20T^{2} \)
73 \( 1 + 2.93e10T + 3.13e20T^{2} \)
79 \( 1 + 1.88e10T + 7.47e20T^{2} \)
83 \( 1 - 2.23e10T + 1.28e21T^{2} \)
89 \( 1 + 9.79e10T + 2.77e21T^{2} \)
97 \( 1 - 2.14e10T + 7.15e21T^{2} \)
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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

−10.02361637493806264929695154500, −8.926003899501891578085303199819, −8.458178838314239892160429784280, −6.86804823996452912113752857055, −6.07170315582238683210131973363, −4.48321832051747685131866754425, −3.75114317752071535073002522266, −3.09986023936051608684577497690, −1.27248707120639021321328990185, 0, 1.27248707120639021321328990185, 3.09986023936051608684577497690, 3.75114317752071535073002522266, 4.48321832051747685131866754425, 6.07170315582238683210131973363, 6.86804823996452912113752857055, 8.458178838314239892160429784280, 8.926003899501891578085303199819, 10.02361637493806264929695154500

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