Properties

Label 2-177-1.1-c11-0-91
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  − 88.8·2-s + 243·3-s + 5.84e3·4-s + 1.13e4·5-s − 2.15e4·6-s + 2.30e4·7-s − 3.36e5·8-s + 5.90e4·9-s − 1.01e6·10-s − 2.40e5·11-s + 1.41e6·12-s − 2.32e6·13-s − 2.04e6·14-s + 2.76e6·15-s + 1.79e7·16-s + 1.04e7·17-s − 5.24e6·18-s − 1.05e7·19-s + 6.65e7·20-s + 5.59e6·21-s + 2.13e7·22-s − 3.94e7·23-s − 8.18e7·24-s + 8.10e7·25-s + 2.06e8·26-s + 1.43e7·27-s + 1.34e8·28-s + ⋯
L(s)  = 1  − 1.96·2-s + 0.577·3-s + 2.85·4-s + 1.63·5-s − 1.13·6-s + 0.517·7-s − 3.63·8-s + 0.333·9-s − 3.20·10-s − 0.450·11-s + 1.64·12-s − 1.73·13-s − 1.01·14-s + 0.941·15-s + 4.28·16-s + 1.78·17-s − 0.654·18-s − 0.973·19-s + 4.65·20-s + 0.298·21-s + 0.883·22-s − 1.27·23-s − 2.09·24-s + 1.65·25-s + 3.41·26-s + 0.192·27-s + 1.47·28-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 \( 1 + 88.8T + 2.04e3T^{2} \)
5 \( 1 - 1.13e4T + 4.88e7T^{2} \)
7 \( 1 - 2.30e4T + 1.97e9T^{2} \)
11 \( 1 + 2.40e5T + 2.85e11T^{2} \)
13 \( 1 + 2.32e6T + 1.79e12T^{2} \)
17 \( 1 - 1.04e7T + 3.42e13T^{2} \)
19 \( 1 + 1.05e7T + 1.16e14T^{2} \)
23 \( 1 + 3.94e7T + 9.52e14T^{2} \)
29 \( 1 + 9.37e6T + 1.22e16T^{2} \)
31 \( 1 + 2.38e7T + 2.54e16T^{2} \)
37 \( 1 - 3.44e8T + 1.77e17T^{2} \)
41 \( 1 + 1.07e9T + 5.50e17T^{2} \)
43 \( 1 - 9.52e8T + 9.29e17T^{2} \)
47 \( 1 + 2.34e9T + 2.47e18T^{2} \)
53 \( 1 + 1.54e9T + 9.26e18T^{2} \)
61 \( 1 - 5.48e9T + 4.35e19T^{2} \)
67 \( 1 + 3.63e9T + 1.22e20T^{2} \)
71 \( 1 + 4.43e9T + 2.31e20T^{2} \)
73 \( 1 + 1.08e10T + 3.13e20T^{2} \)
79 \( 1 + 1.30e10T + 7.47e20T^{2} \)
83 \( 1 + 2.78e10T + 1.28e21T^{2} \)
89 \( 1 - 3.44e10T + 2.77e21T^{2} \)
97 \( 1 - 1.26e11T + 7.15e21T^{2} \)
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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

−9.971271525466922830246406280125, −9.398870717911985039232258387556, −8.221257538257625884164023650511, −7.57307857531527330321756891372, −6.40390973150463781731167166651, −5.34728346316276497165828430406, −2.84441592446991006132670738105, −2.08340185152388931214214548302, −1.42463785645027513281915268896, 0, 1.42463785645027513281915268896, 2.08340185152388931214214548302, 2.84441592446991006132670738105, 5.34728346316276497165828430406, 6.40390973150463781731167166651, 7.57307857531527330321756891372, 8.221257538257625884164023650511, 9.398870717911985039232258387556, 9.971271525466922830246406280125

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