Properties

Label 2-177-1.1-c13-0-84
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  − 169.·2-s − 729·3-s + 2.06e4·4-s + 4.14e4·5-s + 1.23e5·6-s + 2.22e5·7-s − 2.12e6·8-s + 5.31e5·9-s − 7.05e6·10-s − 1.07e7·11-s − 1.50e7·12-s + 2.00e7·13-s − 3.77e7·14-s − 3.02e7·15-s + 1.91e8·16-s + 1.01e8·17-s − 9.02e7·18-s − 3.33e8·19-s + 8.58e8·20-s − 1.62e8·21-s + 1.83e9·22-s + 1.06e9·23-s + 1.54e9·24-s + 5.01e8·25-s − 3.40e9·26-s − 3.87e8·27-s + 4.59e9·28-s + ⋯
L(s)  = 1  − 1.87·2-s − 0.577·3-s + 2.52·4-s + 1.18·5-s + 1.08·6-s + 0.714·7-s − 2.86·8-s + 0.333·9-s − 2.22·10-s − 1.83·11-s − 1.45·12-s + 1.15·13-s − 1.34·14-s − 0.685·15-s + 2.84·16-s + 1.02·17-s − 0.625·18-s − 1.62·19-s + 2.99·20-s − 0.412·21-s + 3.44·22-s + 1.50·23-s + 1.65·24-s + 0.410·25-s − 2.16·26-s − 0.192·27-s + 1.80·28-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 \( 1 + 169.T + 8.19e3T^{2} \)
5 \( 1 - 4.14e4T + 1.22e9T^{2} \)
7 \( 1 - 2.22e5T + 9.68e10T^{2} \)
11 \( 1 + 1.07e7T + 3.45e13T^{2} \)
13 \( 1 - 2.00e7T + 3.02e14T^{2} \)
17 \( 1 - 1.01e8T + 9.90e15T^{2} \)
19 \( 1 + 3.33e8T + 4.20e16T^{2} \)
23 \( 1 - 1.06e9T + 5.04e17T^{2} \)
29 \( 1 - 5.94e9T + 1.02e19T^{2} \)
31 \( 1 + 6.31e9T + 2.44e19T^{2} \)
37 \( 1 + 1.05e10T + 2.43e20T^{2} \)
41 \( 1 + 3.83e10T + 9.25e20T^{2} \)
43 \( 1 - 1.84e10T + 1.71e21T^{2} \)
47 \( 1 - 2.22e10T + 5.46e21T^{2} \)
53 \( 1 - 1.89e10T + 2.60e22T^{2} \)
61 \( 1 - 3.61e11T + 1.61e23T^{2} \)
67 \( 1 + 1.10e12T + 5.48e23T^{2} \)
71 \( 1 + 5.32e11T + 1.16e24T^{2} \)
73 \( 1 - 8.30e11T + 1.67e24T^{2} \)
79 \( 1 + 2.73e12T + 4.66e24T^{2} \)
83 \( 1 - 3.89e12T + 8.87e24T^{2} \)
89 \( 1 + 6.16e12T + 2.19e25T^{2} \)
97 \( 1 + 1.11e13T + 6.73e25T^{2} \)
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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

−10.04121549508203014904021193416, −8.774100329642621225433142002594, −8.158249074451148021592205581468, −7.01880538221701582156459303553, −6.01818004004802432151470133566, −5.16322477056362259240053695512, −2.84842803410853174377621576584, −1.84882266289069097346414280332, −1.11420570534007834789373047447, 0, 1.11420570534007834789373047447, 1.84882266289069097346414280332, 2.84842803410853174377621576584, 5.16322477056362259240053695512, 6.01818004004802432151470133566, 7.01880538221701582156459303553, 8.158249074451148021592205581468, 8.774100329642621225433142002594, 10.04121549508203014904021193416

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