Properties

Label 2-177-1.1-c13-0-89
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  + 143.·2-s − 729·3-s + 1.23e4·4-s − 5.16e4·5-s − 1.04e5·6-s − 5.73e5·7-s + 5.89e5·8-s + 5.31e5·9-s − 7.39e6·10-s + 1.13e7·11-s − 8.97e6·12-s + 1.95e7·13-s − 8.21e7·14-s + 3.76e7·15-s − 1.64e7·16-s + 4.63e7·17-s + 7.60e7·18-s + 3.60e8·19-s − 6.36e8·20-s + 4.18e8·21-s + 1.62e9·22-s − 1.09e9·23-s − 4.29e8·24-s + 1.44e9·25-s + 2.80e9·26-s − 3.87e8·27-s − 7.06e9·28-s + ⋯
L(s)  = 1  + 1.58·2-s − 0.577·3-s + 1.50·4-s − 1.47·5-s − 0.913·6-s − 1.84·7-s + 0.794·8-s + 0.333·9-s − 2.33·10-s + 1.92·11-s − 0.867·12-s + 1.12·13-s − 2.91·14-s + 0.853·15-s − 0.244·16-s + 0.465·17-s + 0.527·18-s + 1.75·19-s − 2.22·20-s + 1.06·21-s + 3.04·22-s − 1.53·23-s − 0.458·24-s + 1.18·25-s + 1.77·26-s − 0.192·27-s − 2.77·28-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 \( 1 - 143.T + 8.19e3T^{2} \)
5 \( 1 + 5.16e4T + 1.22e9T^{2} \)
7 \( 1 + 5.73e5T + 9.68e10T^{2} \)
11 \( 1 - 1.13e7T + 3.45e13T^{2} \)
13 \( 1 - 1.95e7T + 3.02e14T^{2} \)
17 \( 1 - 4.63e7T + 9.90e15T^{2} \)
19 \( 1 - 3.60e8T + 4.20e16T^{2} \)
23 \( 1 + 1.09e9T + 5.04e17T^{2} \)
29 \( 1 - 5.94e9T + 1.02e19T^{2} \)
31 \( 1 + 4.82e9T + 2.44e19T^{2} \)
37 \( 1 + 1.53e10T + 2.43e20T^{2} \)
41 \( 1 + 4.07e9T + 9.25e20T^{2} \)
43 \( 1 + 4.52e8T + 1.71e21T^{2} \)
47 \( 1 + 7.15e10T + 5.46e21T^{2} \)
53 \( 1 - 4.29e10T + 2.60e22T^{2} \)
61 \( 1 - 2.55e11T + 1.61e23T^{2} \)
67 \( 1 + 8.23e11T + 5.48e23T^{2} \)
71 \( 1 + 1.12e12T + 1.16e24T^{2} \)
73 \( 1 + 9.39e11T + 1.67e24T^{2} \)
79 \( 1 + 1.92e12T + 4.66e24T^{2} \)
83 \( 1 + 3.72e11T + 8.87e24T^{2} \)
89 \( 1 - 4.22e12T + 2.19e25T^{2} \)
97 \( 1 - 6.68e11T + 6.73e25T^{2} \)
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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

−10.07098325650781937855476550611, −8.874372494471711230615045847164, −7.23187377125351342616358906307, −6.48337385289413333473856980565, −5.82826118543464741395533318831, −4.34856107173671820714028089097, −3.53296359848193629620948006827, −3.34259721727908366452511829949, −1.15883885365204213649940736855, 0, 1.15883885365204213649940736855, 3.34259721727908366452511829949, 3.53296359848193629620948006827, 4.34856107173671820714028089097, 5.82826118543464741395533318831, 6.48337385289413333473856980565, 7.23187377125351342616358906307, 8.874372494471711230615045847164, 10.07098325650781937855476550611

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