Properties

Label 2-177-1.1-c13-0-9
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  − 121.·2-s + 729·3-s + 6.47e3·4-s − 6.10e4·5-s − 8.82e4·6-s + 2.74e5·7-s + 2.08e5·8-s + 5.31e5·9-s + 7.39e6·10-s − 3.07e6·11-s + 4.71e6·12-s − 2.44e7·13-s − 3.32e7·14-s − 4.45e7·15-s − 7.82e7·16-s + 3.05e7·17-s − 6.43e7·18-s − 2.42e8·19-s − 3.95e8·20-s + 1.99e8·21-s + 3.72e8·22-s + 1.25e8·23-s + 1.51e8·24-s + 2.50e9·25-s + 2.95e9·26-s + 3.87e8·27-s + 1.77e9·28-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.577·3-s + 0.790·4-s − 1.74·5-s − 0.772·6-s + 0.880·7-s + 0.280·8-s + 0.333·9-s + 2.33·10-s − 0.523·11-s + 0.456·12-s − 1.40·13-s − 1.17·14-s − 1.00·15-s − 1.16·16-s + 0.307·17-s − 0.445·18-s − 1.18·19-s − 1.38·20-s + 0.508·21-s + 0.700·22-s + 0.176·23-s + 0.162·24-s + 2.05·25-s + 1.87·26-s + 0.192·27-s + 0.696·28-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

\(L(7)\) \(\approx\) \(0.2417561661\)
\(L(\frac12)\) \(\approx\) \(0.2417561661\)
\(L(\frac{15}{2})\) 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 \( 1 + 121.T + 8.19e3T^{2} \)
5 \( 1 + 6.10e4T + 1.22e9T^{2} \)
7 \( 1 - 2.74e5T + 9.68e10T^{2} \)
11 \( 1 + 3.07e6T + 3.45e13T^{2} \)
13 \( 1 + 2.44e7T + 3.02e14T^{2} \)
17 \( 1 - 3.05e7T + 9.90e15T^{2} \)
19 \( 1 + 2.42e8T + 4.20e16T^{2} \)
23 \( 1 - 1.25e8T + 5.04e17T^{2} \)
29 \( 1 + 1.37e9T + 1.02e19T^{2} \)
31 \( 1 + 3.28e9T + 2.44e19T^{2} \)
37 \( 1 - 9.54e8T + 2.43e20T^{2} \)
41 \( 1 + 2.77e9T + 9.25e20T^{2} \)
43 \( 1 + 1.07e10T + 1.71e21T^{2} \)
47 \( 1 - 4.60e10T + 5.46e21T^{2} \)
53 \( 1 + 8.57e10T + 2.60e22T^{2} \)
61 \( 1 + 8.82e10T + 1.61e23T^{2} \)
67 \( 1 + 3.21e11T + 5.48e23T^{2} \)
71 \( 1 + 3.12e11T + 1.16e24T^{2} \)
73 \( 1 + 1.78e12T + 1.67e24T^{2} \)
79 \( 1 + 3.29e12T + 4.66e24T^{2} \)
83 \( 1 + 2.11e12T + 8.87e24T^{2} \)
89 \( 1 - 5.49e12T + 2.19e25T^{2} \)
97 \( 1 + 1.88e12T + 6.73e25T^{2} \)
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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

−10.28501411120085509853899474966, −9.068697614721738103655657298407, −8.252275903035971404598762374146, −7.68840262669484914202984495256, −7.12556170774841580216137233344, −4.87928385434971344121283962960, −4.11389047487476082405373238545, −2.69928518108817593773537876503, −1.55790086210574724754634362613, −0.25135976023245318076310809044, 0.25135976023245318076310809044, 1.55790086210574724754634362613, 2.69928518108817593773537876503, 4.11389047487476082405373238545, 4.87928385434971344121283962960, 7.12556170774841580216137233344, 7.68840262669484914202984495256, 8.252275903035971404598762374146, 9.068697614721738103655657298407, 10.28501411120085509853899474966

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