Properties

Label 2-177-1.1-c13-0-55
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  + 86.2·2-s − 729·3-s − 750.·4-s − 6.48e4·5-s − 6.28e4·6-s − 9.82e4·7-s − 7.71e5·8-s + 5.31e5·9-s − 5.59e6·10-s − 5.83e6·11-s + 5.47e5·12-s + 2.22e6·13-s − 8.47e6·14-s + 4.72e7·15-s − 6.03e7·16-s + 9.15e7·17-s + 4.58e7·18-s + 1.28e8·19-s + 4.86e7·20-s + 7.16e7·21-s − 5.03e8·22-s + 1.09e9·23-s + 5.62e8·24-s + 2.98e9·25-s + 1.92e8·26-s − 3.87e8·27-s + 7.37e7·28-s + ⋯
L(s)  = 1  + 0.953·2-s − 0.577·3-s − 0.0916·4-s − 1.85·5-s − 0.550·6-s − 0.315·7-s − 1.04·8-s + 0.333·9-s − 1.76·10-s − 0.993·11-s + 0.0529·12-s + 0.128·13-s − 0.300·14-s + 1.07·15-s − 0.899·16-s + 0.920·17-s + 0.317·18-s + 0.626·19-s + 0.170·20-s + 0.182·21-s − 0.946·22-s + 1.54·23-s + 0.600·24-s + 2.44·25-s + 0.122·26-s − 0.192·27-s + 0.0289·28-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 \( 1 - 86.2T + 8.19e3T^{2} \)
5 \( 1 + 6.48e4T + 1.22e9T^{2} \)
7 \( 1 + 9.82e4T + 9.68e10T^{2} \)
11 \( 1 + 5.83e6T + 3.45e13T^{2} \)
13 \( 1 - 2.22e6T + 3.02e14T^{2} \)
17 \( 1 - 9.15e7T + 9.90e15T^{2} \)
19 \( 1 - 1.28e8T + 4.20e16T^{2} \)
23 \( 1 - 1.09e9T + 5.04e17T^{2} \)
29 \( 1 - 5.10e8T + 1.02e19T^{2} \)
31 \( 1 + 2.94e9T + 2.44e19T^{2} \)
37 \( 1 - 2.19e10T + 2.43e20T^{2} \)
41 \( 1 + 5.73e10T + 9.25e20T^{2} \)
43 \( 1 + 6.16e10T + 1.71e21T^{2} \)
47 \( 1 + 4.57e10T + 5.46e21T^{2} \)
53 \( 1 - 2.65e11T + 2.60e22T^{2} \)
61 \( 1 - 1.54e11T + 1.61e23T^{2} \)
67 \( 1 - 2.77e11T + 5.48e23T^{2} \)
71 \( 1 - 8.61e11T + 1.16e24T^{2} \)
73 \( 1 + 1.38e12T + 1.67e24T^{2} \)
79 \( 1 - 1.78e12T + 4.66e24T^{2} \)
83 \( 1 + 3.09e12T + 8.87e24T^{2} \)
89 \( 1 - 3.36e12T + 2.19e25T^{2} \)
97 \( 1 - 4.52e12T + 6.73e25T^{2} \)
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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

−10.05085019204311876597618196987, −8.650271801688706908856709967424, −7.69543103916234518613058927349, −6.73700789423055394451009156699, −5.31716112933554231473096555428, −4.75465532161796468495055925768, −3.58351231971121089232000433652, −3.06306014091965914014892335059, −0.819352943023794591910434239679, 0, 0.819352943023794591910434239679, 3.06306014091965914014892335059, 3.58351231971121089232000433652, 4.75465532161796468495055925768, 5.31716112933554231473096555428, 6.73700789423055394451009156699, 7.69543103916234518613058927349, 8.650271801688706908856709967424, 10.05085019204311876597618196987

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