Properties

Label 2-177-1.1-c13-0-117
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  + 135.·2-s − 729·3-s + 1.02e4·4-s + 1.48e4·5-s − 9.91e4·6-s + 5.65e5·7-s + 2.86e5·8-s + 5.31e5·9-s + 2.02e6·10-s − 4.62e6·11-s − 7.50e6·12-s + 5.26e6·13-s + 7.69e7·14-s − 1.08e7·15-s − 4.54e7·16-s − 1.51e7·17-s + 7.22e7·18-s − 3.97e8·19-s + 1.53e8·20-s − 4.12e8·21-s − 6.28e8·22-s − 4.87e8·23-s − 2.08e8·24-s − 9.99e8·25-s + 7.15e8·26-s − 3.87e8·27-s + 5.82e9·28-s + ⋯
L(s)  = 1  + 1.50·2-s − 0.577·3-s + 1.25·4-s + 0.425·5-s − 0.867·6-s + 1.81·7-s + 0.385·8-s + 0.333·9-s + 0.639·10-s − 0.787·11-s − 0.725·12-s + 0.302·13-s + 2.73·14-s − 0.245·15-s − 0.677·16-s − 0.152·17-s + 0.500·18-s − 1.94·19-s + 0.534·20-s − 1.04·21-s − 1.18·22-s − 0.686·23-s − 0.222·24-s − 0.818·25-s + 0.454·26-s − 0.192·27-s + 2.28·28-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 \( 1 - 135.T + 8.19e3T^{2} \)
5 \( 1 - 1.48e4T + 1.22e9T^{2} \)
7 \( 1 - 5.65e5T + 9.68e10T^{2} \)
11 \( 1 + 4.62e6T + 3.45e13T^{2} \)
13 \( 1 - 5.26e6T + 3.02e14T^{2} \)
17 \( 1 + 1.51e7T + 9.90e15T^{2} \)
19 \( 1 + 3.97e8T + 4.20e16T^{2} \)
23 \( 1 + 4.87e8T + 5.04e17T^{2} \)
29 \( 1 + 1.57e9T + 1.02e19T^{2} \)
31 \( 1 + 1.56e9T + 2.44e19T^{2} \)
37 \( 1 - 1.75e10T + 2.43e20T^{2} \)
41 \( 1 + 6.99e9T + 9.25e20T^{2} \)
43 \( 1 + 1.31e10T + 1.71e21T^{2} \)
47 \( 1 + 5.20e10T + 5.46e21T^{2} \)
53 \( 1 - 6.95e10T + 2.60e22T^{2} \)
61 \( 1 - 2.59e11T + 1.61e23T^{2} \)
67 \( 1 + 5.43e11T + 5.48e23T^{2} \)
71 \( 1 + 6.68e10T + 1.16e24T^{2} \)
73 \( 1 + 1.85e12T + 1.67e24T^{2} \)
79 \( 1 + 3.02e12T + 4.66e24T^{2} \)
83 \( 1 + 1.94e12T + 8.87e24T^{2} \)
89 \( 1 - 8.67e12T + 2.19e25T^{2} \)
97 \( 1 - 1.35e13T + 6.73e25T^{2} \)
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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

−10.34445905821254980656494124494, −8.653423117310710369935109338807, −7.62806030537842001461736601364, −6.26082116883736377676963139841, −5.54572817029689718605538637690, −4.68822411934280246675645619596, −4.03873737417595808246428572317, −2.35740409319428084990693670889, −1.68539916371686865302918481705, 0, 1.68539916371686865302918481705, 2.35740409319428084990693670889, 4.03873737417595808246428572317, 4.68822411934280246675645619596, 5.54572817029689718605538637690, 6.26082116883736377676963139841, 7.62806030537842001461736601364, 8.653423117310710369935109338807, 10.34445905821254980656494124494

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