Properties

Label 2-177-1.1-c13-0-33
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 83.5·2-s + 729·3-s − 1.20e3·4-s + 5.58e4·5-s − 6.09e4·6-s − 5.15e5·7-s + 7.85e5·8-s + 5.31e5·9-s − 4.66e6·10-s + 3.72e6·11-s − 8.81e5·12-s − 9.32e6·13-s + 4.30e7·14-s + 4.07e7·15-s − 5.57e7·16-s − 7.95e6·17-s − 4.44e7·18-s − 6.13e7·19-s − 6.75e7·20-s − 3.75e8·21-s − 3.11e8·22-s + 5.74e8·23-s + 5.72e8·24-s + 1.89e9·25-s + 7.79e8·26-s + 3.87e8·27-s + 6.23e8·28-s + ⋯
L(s)  = 1  − 0.923·2-s + 0.577·3-s − 0.147·4-s + 1.59·5-s − 0.533·6-s − 1.65·7-s + 1.05·8-s + 0.333·9-s − 1.47·10-s + 0.633·11-s − 0.0852·12-s − 0.536·13-s + 1.52·14-s + 0.923·15-s − 0.830·16-s − 0.0799·17-s − 0.307·18-s − 0.299·19-s − 0.235·20-s − 0.955·21-s − 0.585·22-s + 0.808·23-s + 0.611·24-s + 1.55·25-s + 0.494·26-s + 0.192·27-s + 0.244·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(1.582494131\)
\(L(\frac12)\) \(\approx\) \(1.582494131\)
\(L(\frac{15}{2})\) 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 \( 1 + 83.5T + 8.19e3T^{2} \)
5 \( 1 - 5.58e4T + 1.22e9T^{2} \)
7 \( 1 + 5.15e5T + 9.68e10T^{2} \)
11 \( 1 - 3.72e6T + 3.45e13T^{2} \)
13 \( 1 + 9.32e6T + 3.02e14T^{2} \)
17 \( 1 + 7.95e6T + 9.90e15T^{2} \)
19 \( 1 + 6.13e7T + 4.20e16T^{2} \)
23 \( 1 - 5.74e8T + 5.04e17T^{2} \)
29 \( 1 + 1.68e9T + 1.02e19T^{2} \)
31 \( 1 + 7.08e9T + 2.44e19T^{2} \)
37 \( 1 + 2.19e9T + 2.43e20T^{2} \)
41 \( 1 - 7.06e9T + 9.25e20T^{2} \)
43 \( 1 - 4.21e10T + 1.71e21T^{2} \)
47 \( 1 - 3.35e10T + 5.46e21T^{2} \)
53 \( 1 - 1.04e11T + 2.60e22T^{2} \)
61 \( 1 - 1.12e11T + 1.61e23T^{2} \)
67 \( 1 - 5.18e11T + 5.48e23T^{2} \)
71 \( 1 + 7.67e11T + 1.16e24T^{2} \)
73 \( 1 + 6.21e11T + 1.67e24T^{2} \)
79 \( 1 - 2.09e12T + 4.66e24T^{2} \)
83 \( 1 - 8.75e11T + 8.87e24T^{2} \)
89 \( 1 - 4.59e12T + 2.19e25T^{2} \)
97 \( 1 + 4.07e12T + 6.73e25T^{2} \)
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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

−9.915836245891466965410099258128, −9.256246443364565837584990550647, −8.987754345374828074015575664649, −7.33891818419316509333527965740, −6.50580496763658497508188438617, −5.40620223027287071728447935660, −3.90072919448995326965508281797, −2.66117780092369346150963099598, −1.72196746474144329079539193319, −0.60061723644905267186066281175, 0.60061723644905267186066281175, 1.72196746474144329079539193319, 2.66117780092369346150963099598, 3.90072919448995326965508281797, 5.40620223027287071728447935660, 6.50580496763658497508188438617, 7.33891818419316509333527965740, 8.987754345374828074015575664649, 9.256246443364565837584990550647, 9.915836245891466965410099258128

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