Properties

Label 2-177-1.1-c11-0-24
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.89·2-s − 243·3-s − 1.98e3·4-s − 1.40e3·5-s − 1.91e3·6-s + 2.51e4·7-s − 3.18e4·8-s + 5.90e4·9-s − 1.10e4·10-s + 2.51e5·11-s + 4.82e5·12-s + 1.97e6·13-s + 1.98e5·14-s + 3.41e5·15-s + 3.81e6·16-s + 3.84e5·17-s + 4.65e5·18-s − 1.00e7·19-s + 2.79e6·20-s − 6.10e6·21-s + 1.98e6·22-s + 5.12e7·23-s + 7.73e6·24-s − 4.68e7·25-s + 1.56e7·26-s − 1.43e7·27-s − 4.98e7·28-s + ⋯
L(s)  = 1  + 0.174·2-s − 0.577·3-s − 0.969·4-s − 0.201·5-s − 0.100·6-s + 0.564·7-s − 0.343·8-s + 0.333·9-s − 0.0350·10-s + 0.471·11-s + 0.559·12-s + 1.47·13-s + 0.0984·14-s + 0.116·15-s + 0.909·16-s + 0.0656·17-s + 0.0581·18-s − 0.926·19-s + 0.195·20-s − 0.325·21-s + 0.0822·22-s + 1.66·23-s + 0.198·24-s − 0.959·25-s + 0.257·26-s − 0.192·27-s − 0.547·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(1.495863281\)
\(L(\frac12)\) \(\approx\) \(1.495863281\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 + 7.14e8T \)
good2 \( 1 - 7.89T + 2.04e3T^{2} \)
5 \( 1 + 1.40e3T + 4.88e7T^{2} \)
7 \( 1 - 2.51e4T + 1.97e9T^{2} \)
11 \( 1 - 2.51e5T + 2.85e11T^{2} \)
13 \( 1 - 1.97e6T + 1.79e12T^{2} \)
17 \( 1 - 3.84e5T + 3.42e13T^{2} \)
19 \( 1 + 1.00e7T + 1.16e14T^{2} \)
23 \( 1 - 5.12e7T + 9.52e14T^{2} \)
29 \( 1 + 1.79e8T + 1.22e16T^{2} \)
31 \( 1 + 1.54e8T + 2.54e16T^{2} \)
37 \( 1 - 6.49e8T + 1.77e17T^{2} \)
41 \( 1 + 1.77e8T + 5.50e17T^{2} \)
43 \( 1 - 1.32e9T + 9.29e17T^{2} \)
47 \( 1 + 2.57e9T + 2.47e18T^{2} \)
53 \( 1 + 1.18e9T + 9.26e18T^{2} \)
61 \( 1 - 8.62e9T + 4.35e19T^{2} \)
67 \( 1 + 9.42e8T + 1.22e20T^{2} \)
71 \( 1 - 8.96e9T + 2.31e20T^{2} \)
73 \( 1 + 9.87e9T + 3.13e20T^{2} \)
79 \( 1 + 1.71e10T + 7.47e20T^{2} \)
83 \( 1 - 5.86e10T + 1.28e21T^{2} \)
89 \( 1 + 2.87e10T + 2.77e21T^{2} \)
97 \( 1 - 8.39e10T + 7.15e21T^{2} \)
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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

−10.96738070525983641123032325123, −9.543036365991338920390466729427, −8.733836151468912708654703392358, −7.72432736237899597422301109897, −6.31250150252556069735030780863, −5.40450830947861044836662338004, −4.32730561400279496953517257077, −3.54654293494057771689938942282, −1.61054320209426454993316060372, −0.59623679698766838268561880254, 0.59623679698766838268561880254, 1.61054320209426454993316060372, 3.54654293494057771689938942282, 4.32730561400279496953517257077, 5.40450830947861044836662338004, 6.31250150252556069735030780863, 7.72432736237899597422301109897, 8.733836151468912708654703392358, 9.543036365991338920390466729427, 10.96738070525983641123032325123

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