Properties

Label 2-177-1.1-c13-0-46
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  + 52.9·2-s − 729·3-s − 5.38e3·4-s − 3.31e4·5-s − 3.85e4·6-s − 4.13e5·7-s − 7.18e5·8-s + 5.31e5·9-s − 1.75e6·10-s + 2.37e6·11-s + 3.92e6·12-s − 1.44e7·13-s − 2.18e7·14-s + 2.41e7·15-s + 6.09e6·16-s − 9.80e7·17-s + 2.81e7·18-s + 2.31e8·19-s + 1.78e8·20-s + 3.01e8·21-s + 1.25e8·22-s + 7.61e8·23-s + 5.24e8·24-s − 1.21e8·25-s − 7.66e8·26-s − 3.87e8·27-s + 2.22e9·28-s + ⋯
L(s)  = 1  + 0.584·2-s − 0.577·3-s − 0.657·4-s − 0.948·5-s − 0.337·6-s − 1.32·7-s − 0.969·8-s + 0.333·9-s − 0.554·10-s + 0.403·11-s + 0.379·12-s − 0.832·13-s − 0.776·14-s + 0.547·15-s + 0.0908·16-s − 0.985·17-s + 0.194·18-s + 1.13·19-s + 0.624·20-s + 0.766·21-s + 0.235·22-s + 1.07·23-s + 0.559·24-s − 0.0997·25-s − 0.486·26-s − 0.192·27-s + 0.873·28-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 \( 1 - 52.9T + 8.19e3T^{2} \)
5 \( 1 + 3.31e4T + 1.22e9T^{2} \)
7 \( 1 + 4.13e5T + 9.68e10T^{2} \)
11 \( 1 - 2.37e6T + 3.45e13T^{2} \)
13 \( 1 + 1.44e7T + 3.02e14T^{2} \)
17 \( 1 + 9.80e7T + 9.90e15T^{2} \)
19 \( 1 - 2.31e8T + 4.20e16T^{2} \)
23 \( 1 - 7.61e8T + 5.04e17T^{2} \)
29 \( 1 + 2.21e9T + 1.02e19T^{2} \)
31 \( 1 - 6.65e8T + 2.44e19T^{2} \)
37 \( 1 - 3.56e9T + 2.43e20T^{2} \)
41 \( 1 - 5.94e10T + 9.25e20T^{2} \)
43 \( 1 - 3.00e9T + 1.71e21T^{2} \)
47 \( 1 - 7.04e10T + 5.46e21T^{2} \)
53 \( 1 + 2.38e11T + 2.60e22T^{2} \)
61 \( 1 - 4.48e11T + 1.61e23T^{2} \)
67 \( 1 + 1.96e11T + 5.48e23T^{2} \)
71 \( 1 + 4.00e11T + 1.16e24T^{2} \)
73 \( 1 + 2.04e11T + 1.67e24T^{2} \)
79 \( 1 - 1.96e12T + 4.66e24T^{2} \)
83 \( 1 - 4.20e11T + 8.87e24T^{2} \)
89 \( 1 - 4.35e11T + 2.19e25T^{2} \)
97 \( 1 + 6.74e12T + 6.73e25T^{2} \)
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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

−9.652588184537143780420006869217, −9.139614166881172271222515904447, −7.62419080476729346996757796679, −6.67033404598368493105382215259, −5.62784266809255973450324763293, −4.55336437096867093311163353856, −3.74818627487371735768268992963, −2.80172076546955243393319066483, −0.75000679266519900201010592475, 0, 0.75000679266519900201010592475, 2.80172076546955243393319066483, 3.74818627487371735768268992963, 4.55336437096867093311163353856, 5.62784266809255973450324763293, 6.67033404598368493105382215259, 7.62419080476729346996757796679, 9.139614166881172271222515904447, 9.652588184537143780420006869217

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