Properties

Label 2-177-1.1-c13-0-108
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  − 49.4·2-s + 729·3-s − 5.74e3·4-s + 4.51e4·5-s − 3.60e4·6-s + 1.16e5·7-s + 6.89e5·8-s + 5.31e5·9-s − 2.23e6·10-s + 2.81e6·11-s − 4.18e6·12-s − 3.96e6·13-s − 5.74e6·14-s + 3.29e7·15-s + 1.29e7·16-s − 9.19e7·17-s − 2.62e7·18-s + 2.77e8·19-s − 2.59e8·20-s + 8.45e7·21-s − 1.39e8·22-s − 6.95e8·23-s + 5.02e8·24-s + 8.19e8·25-s + 1.96e8·26-s + 3.87e8·27-s − 6.66e8·28-s + ⋯
L(s)  = 1  − 0.546·2-s + 0.577·3-s − 0.701·4-s + 1.29·5-s − 0.315·6-s + 0.372·7-s + 0.929·8-s + 0.333·9-s − 0.706·10-s + 0.478·11-s − 0.404·12-s − 0.227·13-s − 0.203·14-s + 0.746·15-s + 0.192·16-s − 0.924·17-s − 0.182·18-s + 1.35·19-s − 0.906·20-s + 0.215·21-s − 0.261·22-s − 0.980·23-s + 0.536·24-s + 0.671·25-s + 0.124·26-s + 0.192·27-s − 0.261·28-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 \( 1 + 49.4T + 8.19e3T^{2} \)
5 \( 1 - 4.51e4T + 1.22e9T^{2} \)
7 \( 1 - 1.16e5T + 9.68e10T^{2} \)
11 \( 1 - 2.81e6T + 3.45e13T^{2} \)
13 \( 1 + 3.96e6T + 3.02e14T^{2} \)
17 \( 1 + 9.19e7T + 9.90e15T^{2} \)
19 \( 1 - 2.77e8T + 4.20e16T^{2} \)
23 \( 1 + 6.95e8T + 5.04e17T^{2} \)
29 \( 1 + 4.72e9T + 1.02e19T^{2} \)
31 \( 1 - 1.19e9T + 2.44e19T^{2} \)
37 \( 1 + 1.10e10T + 2.43e20T^{2} \)
41 \( 1 + 4.13e10T + 9.25e20T^{2} \)
43 \( 1 + 4.95e10T + 1.71e21T^{2} \)
47 \( 1 - 1.54e10T + 5.46e21T^{2} \)
53 \( 1 - 1.00e11T + 2.60e22T^{2} \)
61 \( 1 + 4.41e11T + 1.61e23T^{2} \)
67 \( 1 + 1.19e12T + 5.48e23T^{2} \)
71 \( 1 - 2.02e12T + 1.16e24T^{2} \)
73 \( 1 + 1.19e11T + 1.67e24T^{2} \)
79 \( 1 + 1.71e11T + 4.66e24T^{2} \)
83 \( 1 + 5.24e12T + 8.87e24T^{2} \)
89 \( 1 - 2.52e12T + 2.19e25T^{2} \)
97 \( 1 + 6.11e12T + 6.73e25T^{2} \)
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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

−9.644079969714802742070866290272, −9.051653553220048848644207070495, −8.103850760335112939514000804575, −6.99960496128139279160721266295, −5.66850451607522504132902032885, −4.71152510072627389725190380009, −3.48422563887987199362022559215, −1.99177894097185104141602656771, −1.42271553395366380535612161999, 0, 1.42271553395366380535612161999, 1.99177894097185104141602656771, 3.48422563887987199362022559215, 4.71152510072627389725190380009, 5.66850451607522504132902032885, 6.99960496128139279160721266295, 8.103850760335112939514000804575, 9.051653553220048848644207070495, 9.644079969714802742070866290272

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