Properties

Label 2-177-1.1-c13-0-60
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  − 41.9·2-s + 729·3-s − 6.43e3·4-s − 1.04e4·5-s − 3.05e4·6-s − 4.84e5·7-s + 6.13e5·8-s + 5.31e5·9-s + 4.37e5·10-s − 5.22e6·11-s − 4.69e6·12-s − 2.63e7·13-s + 2.03e7·14-s − 7.60e6·15-s + 2.69e7·16-s + 5.55e7·17-s − 2.22e7·18-s + 2.91e8·19-s + 6.70e7·20-s − 3.53e8·21-s + 2.19e8·22-s + 4.45e8·23-s + 4.47e8·24-s − 1.11e9·25-s + 1.10e9·26-s + 3.87e8·27-s + 3.11e9·28-s + ⋯
L(s)  = 1  − 0.463·2-s + 0.577·3-s − 0.785·4-s − 0.298·5-s − 0.267·6-s − 1.55·7-s + 0.827·8-s + 0.333·9-s + 0.138·10-s − 0.889·11-s − 0.453·12-s − 1.51·13-s + 0.721·14-s − 0.172·15-s + 0.402·16-s + 0.557·17-s − 0.154·18-s + 1.42·19-s + 0.234·20-s − 0.899·21-s + 0.411·22-s + 0.627·23-s + 0.477·24-s − 0.910·25-s + 0.700·26-s + 0.192·27-s + 1.22·28-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 \( 1 + 41.9T + 8.19e3T^{2} \)
5 \( 1 + 1.04e4T + 1.22e9T^{2} \)
7 \( 1 + 4.84e5T + 9.68e10T^{2} \)
11 \( 1 + 5.22e6T + 3.45e13T^{2} \)
13 \( 1 + 2.63e7T + 3.02e14T^{2} \)
17 \( 1 - 5.55e7T + 9.90e15T^{2} \)
19 \( 1 - 2.91e8T + 4.20e16T^{2} \)
23 \( 1 - 4.45e8T + 5.04e17T^{2} \)
29 \( 1 - 1.14e9T + 1.02e19T^{2} \)
31 \( 1 - 1.75e9T + 2.44e19T^{2} \)
37 \( 1 - 1.94e10T + 2.43e20T^{2} \)
41 \( 1 - 4.10e10T + 9.25e20T^{2} \)
43 \( 1 + 6.68e10T + 1.71e21T^{2} \)
47 \( 1 - 1.07e11T + 5.46e21T^{2} \)
53 \( 1 - 1.14e11T + 2.60e22T^{2} \)
61 \( 1 + 3.36e11T + 1.61e23T^{2} \)
67 \( 1 + 2.42e11T + 5.48e23T^{2} \)
71 \( 1 - 4.15e11T + 1.16e24T^{2} \)
73 \( 1 + 1.88e12T + 1.67e24T^{2} \)
79 \( 1 + 1.69e12T + 4.66e24T^{2} \)
83 \( 1 + 1.99e12T + 8.87e24T^{2} \)
89 \( 1 - 2.92e12T + 2.19e25T^{2} \)
97 \( 1 - 1.31e13T + 6.73e25T^{2} \)
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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

−9.796920782719106096999455370184, −9.072427305938269695484505733571, −7.77177637271597653583500978671, −7.28471290871491587748129577695, −5.67366468277017246837582601048, −4.56502104170194884176881758988, −3.35893630561598420016766900154, −2.59269456779485830252659267675, −0.858215432629386744512017466509, 0, 0.858215432629386744512017466509, 2.59269456779485830252659267675, 3.35893630561598420016766900154, 4.56502104170194884176881758988, 5.67366468277017246837582601048, 7.28471290871491587748129577695, 7.77177637271597653583500978671, 9.072427305938269695484505733571, 9.796920782719106096999455370184

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