Properties

Label 2-177-1.1-c13-0-75
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

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 13.8·2-s − 729·3-s − 7.99e3·4-s + 2.43e4·5-s − 1.01e4·6-s − 3.88e4·7-s − 2.24e5·8-s + 5.31e5·9-s + 3.38e5·10-s + 1.36e6·11-s + 5.83e6·12-s − 1.15e7·13-s − 5.39e5·14-s − 1.77e7·15-s + 6.24e7·16-s + 5.54e7·17-s + 7.38e6·18-s − 2.71e8·19-s − 1.94e8·20-s + 2.82e7·21-s + 1.89e7·22-s + 4.08e8·23-s + 1.64e8·24-s − 6.27e8·25-s − 1.61e8·26-s − 3.87e8·27-s + 3.10e8·28-s + ⋯
L(s)  = 1  + 0.153·2-s − 0.577·3-s − 0.976·4-s + 0.697·5-s − 0.0886·6-s − 0.124·7-s − 0.303·8-s + 0.333·9-s + 0.107·10-s + 0.231·11-s + 0.563·12-s − 0.666·13-s − 0.0191·14-s − 0.402·15-s + 0.929·16-s + 0.556·17-s + 0.0511·18-s − 1.32·19-s − 0.680·20-s + 0.0720·21-s + 0.0355·22-s + 0.575·23-s + 0.175·24-s − 0.514·25-s − 0.102·26-s − 0.192·27-s + 0.121·28-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 \( 1 - 13.8T + 8.19e3T^{2} \)
5 \( 1 - 2.43e4T + 1.22e9T^{2} \)
7 \( 1 + 3.88e4T + 9.68e10T^{2} \)
11 \( 1 - 1.36e6T + 3.45e13T^{2} \)
13 \( 1 + 1.15e7T + 3.02e14T^{2} \)
17 \( 1 - 5.54e7T + 9.90e15T^{2} \)
19 \( 1 + 2.71e8T + 4.20e16T^{2} \)
23 \( 1 - 4.08e8T + 5.04e17T^{2} \)
29 \( 1 + 1.59e9T + 1.02e19T^{2} \)
31 \( 1 - 4.27e9T + 2.44e19T^{2} \)
37 \( 1 - 2.54e9T + 2.43e20T^{2} \)
41 \( 1 + 2.20e10T + 9.25e20T^{2} \)
43 \( 1 - 5.67e10T + 1.71e21T^{2} \)
47 \( 1 - 5.63e10T + 5.46e21T^{2} \)
53 \( 1 - 9.18e10T + 2.60e22T^{2} \)
61 \( 1 - 3.59e11T + 1.61e23T^{2} \)
67 \( 1 - 4.39e10T + 5.48e23T^{2} \)
71 \( 1 - 1.08e12T + 1.16e24T^{2} \)
73 \( 1 + 1.32e12T + 1.67e24T^{2} \)
79 \( 1 - 3.85e12T + 4.66e24T^{2} \)
83 \( 1 - 2.36e12T + 8.87e24T^{2} \)
89 \( 1 - 6.93e12T + 2.19e25T^{2} \)
97 \( 1 - 7.40e12T + 6.73e25T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.825601384116861886420714523137, −9.068736262351196402382103789624, −7.889522394796837834299638930774, −6.54521786345662760162565280473, −5.64128295227301570765520011000, −4.77799490530766575607016778992, −3.77433065642647430875901095861, −2.32111168741963114096624830571, −1.01611019776428907388216151124, 0, 1.01611019776428907388216151124, 2.32111168741963114096624830571, 3.77433065642647430875901095861, 4.77799490530766575607016778992, 5.64128295227301570765520011000, 6.54521786345662760162565280473, 7.889522394796837834299638930774, 9.068736262351196402382103789624, 9.825601384116861886420714523137

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