Properties

Label 2-177-1.1-c13-0-104
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  + 112.·2-s − 729·3-s + 4.53e3·4-s + 2.84e4·5-s − 8.22e4·6-s − 2.24e5·7-s − 4.12e5·8-s + 5.31e5·9-s + 3.20e6·10-s + 5.94e6·11-s − 3.30e6·12-s − 2.08e7·13-s − 2.53e7·14-s − 2.07e7·15-s − 8.36e7·16-s + 1.51e8·17-s + 5.99e7·18-s − 1.15e7·19-s + 1.28e8·20-s + 1.63e8·21-s + 6.70e8·22-s + 1.14e9·23-s + 3.00e8·24-s − 4.11e8·25-s − 2.35e9·26-s − 3.87e8·27-s − 1.01e9·28-s + ⋯
L(s)  = 1  + 1.24·2-s − 0.577·3-s + 0.553·4-s + 0.814·5-s − 0.719·6-s − 0.720·7-s − 0.556·8-s + 0.333·9-s + 1.01·10-s + 1.01·11-s − 0.319·12-s − 1.19·13-s − 0.898·14-s − 0.470·15-s − 1.24·16-s + 1.51·17-s + 0.415·18-s − 0.0565·19-s + 0.450·20-s + 0.416·21-s + 1.26·22-s + 1.60·23-s + 0.321·24-s − 0.336·25-s − 1.49·26-s − 0.192·27-s − 0.399·28-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 \( 1 - 112.T + 8.19e3T^{2} \)
5 \( 1 - 2.84e4T + 1.22e9T^{2} \)
7 \( 1 + 2.24e5T + 9.68e10T^{2} \)
11 \( 1 - 5.94e6T + 3.45e13T^{2} \)
13 \( 1 + 2.08e7T + 3.02e14T^{2} \)
17 \( 1 - 1.51e8T + 9.90e15T^{2} \)
19 \( 1 + 1.15e7T + 4.20e16T^{2} \)
23 \( 1 - 1.14e9T + 5.04e17T^{2} \)
29 \( 1 + 1.30e8T + 1.02e19T^{2} \)
31 \( 1 + 8.79e9T + 2.44e19T^{2} \)
37 \( 1 - 2.75e10T + 2.43e20T^{2} \)
41 \( 1 + 5.59e8T + 9.25e20T^{2} \)
43 \( 1 - 1.45e10T + 1.71e21T^{2} \)
47 \( 1 + 5.89e10T + 5.46e21T^{2} \)
53 \( 1 + 2.21e10T + 2.60e22T^{2} \)
61 \( 1 + 5.30e11T + 1.61e23T^{2} \)
67 \( 1 + 4.06e11T + 5.48e23T^{2} \)
71 \( 1 + 1.24e12T + 1.16e24T^{2} \)
73 \( 1 - 2.17e12T + 1.67e24T^{2} \)
79 \( 1 + 3.22e12T + 4.66e24T^{2} \)
83 \( 1 - 5.27e12T + 8.87e24T^{2} \)
89 \( 1 + 3.76e11T + 2.19e25T^{2} \)
97 \( 1 + 8.14e12T + 6.73e25T^{2} \)
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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

−9.689465499239806549308928351291, −9.330443248949308223132789600074, −7.35895584698779103503367428105, −6.36588292236417122314185889960, −5.64404162517254491542095228123, −4.83215205464810337974178414101, −3.65874127865053183012501455867, −2.71505729608588780292699002392, −1.33418984134318495104325129895, 0, 1.33418984134318495104325129895, 2.71505729608588780292699002392, 3.65874127865053183012501455867, 4.83215205464810337974178414101, 5.64404162517254491542095228123, 6.36588292236417122314185889960, 7.35895584698779103503367428105, 9.330443248949308223132789600074, 9.689465499239806549308928351291

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