Properties

Label 2-177-1.1-c13-0-112
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  + 76.7·2-s + 729·3-s − 2.29e3·4-s + 8.61e3·5-s + 5.59e4·6-s + 3.43e5·7-s − 8.05e5·8-s + 5.31e5·9-s + 6.61e5·10-s + 2.10e6·11-s − 1.67e6·12-s + 1.64e7·13-s + 2.63e7·14-s + 6.27e6·15-s − 4.30e7·16-s − 1.54e8·17-s + 4.08e7·18-s − 1.40e8·19-s − 1.97e7·20-s + 2.50e8·21-s + 1.61e8·22-s − 7.50e8·23-s − 5.87e8·24-s − 1.14e9·25-s + 1.26e9·26-s + 3.87e8·27-s − 7.88e8·28-s + ⋯
L(s)  = 1  + 0.848·2-s + 0.577·3-s − 0.280·4-s + 0.246·5-s + 0.489·6-s + 1.10·7-s − 1.08·8-s + 0.333·9-s + 0.209·10-s + 0.358·11-s − 0.161·12-s + 0.943·13-s + 0.936·14-s + 0.142·15-s − 0.641·16-s − 1.54·17-s + 0.282·18-s − 0.685·19-s − 0.0690·20-s + 0.637·21-s + 0.304·22-s − 1.05·23-s − 0.627·24-s − 0.939·25-s + 0.800·26-s + 0.192·27-s − 0.309·28-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 \( 1 - 76.7T + 8.19e3T^{2} \)
5 \( 1 - 8.61e3T + 1.22e9T^{2} \)
7 \( 1 - 3.43e5T + 9.68e10T^{2} \)
11 \( 1 - 2.10e6T + 3.45e13T^{2} \)
13 \( 1 - 1.64e7T + 3.02e14T^{2} \)
17 \( 1 + 1.54e8T + 9.90e15T^{2} \)
19 \( 1 + 1.40e8T + 4.20e16T^{2} \)
23 \( 1 + 7.50e8T + 5.04e17T^{2} \)
29 \( 1 + 5.30e9T + 1.02e19T^{2} \)
31 \( 1 - 6.09e9T + 2.44e19T^{2} \)
37 \( 1 - 1.01e10T + 2.43e20T^{2} \)
41 \( 1 + 2.18e8T + 9.25e20T^{2} \)
43 \( 1 - 6.90e9T + 1.71e21T^{2} \)
47 \( 1 + 7.05e10T + 5.46e21T^{2} \)
53 \( 1 - 1.26e11T + 2.60e22T^{2} \)
61 \( 1 - 3.47e11T + 1.61e23T^{2} \)
67 \( 1 - 7.19e11T + 5.48e23T^{2} \)
71 \( 1 + 1.79e12T + 1.16e24T^{2} \)
73 \( 1 + 6.59e11T + 1.67e24T^{2} \)
79 \( 1 - 6.24e11T + 4.66e24T^{2} \)
83 \( 1 + 3.14e12T + 8.87e24T^{2} \)
89 \( 1 + 5.10e12T + 2.19e25T^{2} \)
97 \( 1 - 2.84e12T + 6.73e25T^{2} \)
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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

−9.695291447754338884558020572530, −8.718384162981136997541817614066, −8.083543873313269357519091234670, −6.54897001559769308697238505260, −5.60695871538086574948828362958, −4.35073985057231355554318383997, −3.93102823430236647134910702629, −2.44795636238316281726159660457, −1.54008737578427757406804467508, 0, 1.54008737578427757406804467508, 2.44795636238316281726159660457, 3.93102823430236647134910702629, 4.35073985057231355554318383997, 5.60695871538086574948828362958, 6.54897001559769308697238505260, 8.083543873313269357519091234670, 8.718384162981136997541817614066, 9.695291447754338884558020572530

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