Properties

Label 2-177-1.1-c13-0-113
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  + 177.·2-s + 729·3-s + 2.32e4·4-s + 3.79e4·5-s + 1.29e5·6-s + 2.36e5·7-s + 2.66e6·8-s + 5.31e5·9-s + 6.73e6·10-s + 1.90e6·11-s + 1.69e7·12-s − 1.09e7·13-s + 4.19e7·14-s + 2.76e7·15-s + 2.82e8·16-s + 1.36e8·17-s + 9.42e7·18-s − 3.07e8·19-s + 8.82e8·20-s + 1.72e8·21-s + 3.37e8·22-s + 5.31e8·23-s + 1.94e9·24-s + 2.22e8·25-s − 1.93e9·26-s + 3.87e8·27-s + 5.49e9·28-s + ⋯
L(s)  = 1  + 1.95·2-s + 0.577·3-s + 2.83·4-s + 1.08·5-s + 1.13·6-s + 0.759·7-s + 3.59·8-s + 0.333·9-s + 2.12·10-s + 0.324·11-s + 1.63·12-s − 0.626·13-s + 1.48·14-s + 0.627·15-s + 4.21·16-s + 1.36·17-s + 0.652·18-s − 1.49·19-s + 3.08·20-s + 0.438·21-s + 0.635·22-s + 0.748·23-s + 2.07·24-s + 0.182·25-s − 1.22·26-s + 0.192·27-s + 2.15·28-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

\(L(7)\) \(\approx\) \(16.67719948\)
\(L(\frac12)\) \(\approx\) \(16.67719948\)
\(L(\frac{15}{2})\) 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 \( 1 - 177.T + 8.19e3T^{2} \)
5 \( 1 - 3.79e4T + 1.22e9T^{2} \)
7 \( 1 - 2.36e5T + 9.68e10T^{2} \)
11 \( 1 - 1.90e6T + 3.45e13T^{2} \)
13 \( 1 + 1.09e7T + 3.02e14T^{2} \)
17 \( 1 - 1.36e8T + 9.90e15T^{2} \)
19 \( 1 + 3.07e8T + 4.20e16T^{2} \)
23 \( 1 - 5.31e8T + 5.04e17T^{2} \)
29 \( 1 + 3.71e9T + 1.02e19T^{2} \)
31 \( 1 + 8.66e9T + 2.44e19T^{2} \)
37 \( 1 - 2.18e10T + 2.43e20T^{2} \)
41 \( 1 - 5.66e9T + 9.25e20T^{2} \)
43 \( 1 + 7.28e10T + 1.71e21T^{2} \)
47 \( 1 - 5.64e10T + 5.46e21T^{2} \)
53 \( 1 + 2.16e11T + 2.60e22T^{2} \)
61 \( 1 - 1.58e11T + 1.61e23T^{2} \)
67 \( 1 - 3.98e11T + 5.48e23T^{2} \)
71 \( 1 + 9.16e11T + 1.16e24T^{2} \)
73 \( 1 - 6.04e11T + 1.67e24T^{2} \)
79 \( 1 - 5.26e11T + 4.66e24T^{2} \)
83 \( 1 + 3.86e12T + 8.87e24T^{2} \)
89 \( 1 + 7.64e12T + 2.19e25T^{2} \)
97 \( 1 - 9.71e12T + 6.73e25T^{2} \)
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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

−10.66074267646188122184331888968, −9.573994673014499847326934284331, −7.953522140730149297890735038663, −6.98781085840492608596152142966, −5.89832464884810924122090303859, −5.16730705726993251960514511441, −4.16938000517343843411404844237, −3.11534190034159985338345902499, −2.05562658000919056986425242825, −1.55004924347812682259768381222, 1.55004924347812682259768381222, 2.05562658000919056986425242825, 3.11534190034159985338345902499, 4.16938000517343843411404844237, 5.16730705726993251960514511441, 5.89832464884810924122090303859, 6.98781085840492608596152142966, 7.953522140730149297890735038663, 9.573994673014499847326934284331, 10.66074267646188122184331888968

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