Properties

Label 2-177-1.1-c13-0-63
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  + 43.5·2-s + 729·3-s − 6.29e3·4-s − 5.38e4·5-s + 3.17e4·6-s − 6.00e5·7-s − 6.31e5·8-s + 5.31e5·9-s − 2.34e6·10-s + 4.53e6·11-s − 4.58e6·12-s + 1.89e7·13-s − 2.61e7·14-s − 3.92e7·15-s + 2.40e7·16-s + 5.77e7·17-s + 2.31e7·18-s − 1.29e7·19-s + 3.38e8·20-s − 4.37e8·21-s + 1.97e8·22-s + 5.60e8·23-s − 4.60e8·24-s + 1.67e9·25-s + 8.27e8·26-s + 3.87e8·27-s + 3.77e9·28-s + ⋯
L(s)  = 1  + 0.481·2-s + 0.577·3-s − 0.768·4-s − 1.54·5-s + 0.277·6-s − 1.92·7-s − 0.851·8-s + 0.333·9-s − 0.741·10-s + 0.772·11-s − 0.443·12-s + 1.09·13-s − 0.928·14-s − 0.889·15-s + 0.358·16-s + 0.580·17-s + 0.160·18-s − 0.0630·19-s + 1.18·20-s − 1.11·21-s + 0.371·22-s + 0.789·23-s − 0.491·24-s + 1.37·25-s + 0.525·26-s + 0.192·27-s + 1.48·28-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 \( 1 - 43.5T + 8.19e3T^{2} \)
5 \( 1 + 5.38e4T + 1.22e9T^{2} \)
7 \( 1 + 6.00e5T + 9.68e10T^{2} \)
11 \( 1 - 4.53e6T + 3.45e13T^{2} \)
13 \( 1 - 1.89e7T + 3.02e14T^{2} \)
17 \( 1 - 5.77e7T + 9.90e15T^{2} \)
19 \( 1 + 1.29e7T + 4.20e16T^{2} \)
23 \( 1 - 5.60e8T + 5.04e17T^{2} \)
29 \( 1 + 4.37e9T + 1.02e19T^{2} \)
31 \( 1 + 7.07e9T + 2.44e19T^{2} \)
37 \( 1 - 1.95e10T + 2.43e20T^{2} \)
41 \( 1 - 1.43e10T + 9.25e20T^{2} \)
43 \( 1 - 1.26e10T + 1.71e21T^{2} \)
47 \( 1 + 2.13e10T + 5.46e21T^{2} \)
53 \( 1 - 1.08e11T + 2.60e22T^{2} \)
61 \( 1 + 6.15e11T + 1.61e23T^{2} \)
67 \( 1 - 2.30e11T + 5.48e23T^{2} \)
71 \( 1 + 4.88e10T + 1.16e24T^{2} \)
73 \( 1 - 2.23e12T + 1.67e24T^{2} \)
79 \( 1 + 3.41e12T + 4.66e24T^{2} \)
83 \( 1 - 2.93e12T + 8.87e24T^{2} \)
89 \( 1 - 4.55e12T + 2.19e25T^{2} \)
97 \( 1 + 8.26e12T + 6.73e25T^{2} \)
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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

−9.429710614936970497327996084571, −9.053971866194178128510190599285, −7.87861874469178273770584331926, −6.82070510116108564331708023073, −5.74608514387958499220950909854, −4.07558070213356719641529903068, −3.68418377527729489598815804381, −3.05268615533647666437304192994, −0.898884417959466176718457264691, 0, 0.898884417959466176718457264691, 3.05268615533647666437304192994, 3.68418377527729489598815804381, 4.07558070213356719641529903068, 5.74608514387958499220950909854, 6.82070510116108564331708023073, 7.87861874469178273770584331926, 9.053971866194178128510190599285, 9.429710614936970497327996084571

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