Properties

Label 2-177-1.1-c13-0-76
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  − 108.·2-s + 729·3-s + 3.62e3·4-s − 1.56e4·5-s − 7.92e4·6-s − 2.22e5·7-s + 4.96e5·8-s + 5.31e5·9-s + 1.70e6·10-s + 5.89e6·11-s + 2.64e6·12-s + 5.02e6·13-s + 2.42e7·14-s − 1.14e7·15-s − 8.36e7·16-s + 1.53e8·17-s − 5.77e7·18-s − 3.69e8·19-s − 5.68e7·20-s − 1.62e8·21-s − 6.40e8·22-s − 5.84e8·23-s + 3.61e8·24-s − 9.74e8·25-s − 5.46e8·26-s + 3.87e8·27-s − 8.08e8·28-s + ⋯
L(s)  = 1  − 1.20·2-s + 0.577·3-s + 0.442·4-s − 0.448·5-s − 0.693·6-s − 0.716·7-s + 0.669·8-s + 0.333·9-s + 0.538·10-s + 1.00·11-s + 0.255·12-s + 0.288·13-s + 0.860·14-s − 0.259·15-s − 1.24·16-s + 1.53·17-s − 0.400·18-s − 1.80·19-s − 0.198·20-s − 0.413·21-s − 1.20·22-s − 0.822·23-s + 0.386·24-s − 0.798·25-s − 0.346·26-s + 0.192·27-s − 0.316·28-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 \( 1 + 108.T + 8.19e3T^{2} \)
5 \( 1 + 1.56e4T + 1.22e9T^{2} \)
7 \( 1 + 2.22e5T + 9.68e10T^{2} \)
11 \( 1 - 5.89e6T + 3.45e13T^{2} \)
13 \( 1 - 5.02e6T + 3.02e14T^{2} \)
17 \( 1 - 1.53e8T + 9.90e15T^{2} \)
19 \( 1 + 3.69e8T + 4.20e16T^{2} \)
23 \( 1 + 5.84e8T + 5.04e17T^{2} \)
29 \( 1 - 4.10e9T + 1.02e19T^{2} \)
31 \( 1 + 3.52e9T + 2.44e19T^{2} \)
37 \( 1 - 1.57e10T + 2.43e20T^{2} \)
41 \( 1 + 4.03e10T + 9.25e20T^{2} \)
43 \( 1 + 2.10e10T + 1.71e21T^{2} \)
47 \( 1 - 6.19e10T + 5.46e21T^{2} \)
53 \( 1 - 2.85e11T + 2.60e22T^{2} \)
61 \( 1 + 1.72e11T + 1.61e23T^{2} \)
67 \( 1 - 4.13e10T + 5.48e23T^{2} \)
71 \( 1 + 6.91e11T + 1.16e24T^{2} \)
73 \( 1 - 1.67e12T + 1.67e24T^{2} \)
79 \( 1 + 9.79e11T + 4.66e24T^{2} \)
83 \( 1 + 4.95e12T + 8.87e24T^{2} \)
89 \( 1 - 1.48e11T + 2.19e25T^{2} \)
97 \( 1 - 7.18e11T + 6.73e25T^{2} \)
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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

−9.799180918070070331720201135128, −8.755193377278841430103614121601, −8.174411164877817295592333983722, −7.15086688996526115367082037743, −6.13032555923607392277491588876, −4.31510935628405947602575639353, −3.51356577481106259898810654862, −2.06919029858374264968213462354, −1.02570391004201724852668406305, 0, 1.02570391004201724852668406305, 2.06919029858374264968213462354, 3.51356577481106259898810654862, 4.31510935628405947602575639353, 6.13032555923607392277491588876, 7.15086688996526115367082037743, 8.174411164877817295592333983722, 8.755193377278841430103614121601, 9.799180918070070331720201135128

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