Properties

Label 2-177-1.1-c11-0-84
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  + 79.9·2-s + 243·3-s + 4.34e3·4-s + 3.57e3·5-s + 1.94e4·6-s + 3.42e4·7-s + 1.84e5·8-s + 5.90e4·9-s + 2.85e5·10-s + 5.61e5·11-s + 1.05e6·12-s − 8.41e5·13-s + 2.74e6·14-s + 8.68e5·15-s + 5.81e6·16-s − 1.34e6·17-s + 4.72e6·18-s − 1.30e6·19-s + 1.55e7·20-s + 8.33e6·21-s + 4.49e7·22-s + 5.35e7·23-s + 4.47e7·24-s − 3.60e7·25-s − 6.73e7·26-s + 1.43e7·27-s + 1.49e8·28-s + ⋯
L(s)  = 1  + 1.76·2-s + 0.577·3-s + 2.12·4-s + 0.511·5-s + 1.02·6-s + 0.771·7-s + 1.98·8-s + 0.333·9-s + 0.903·10-s + 1.05·11-s + 1.22·12-s − 0.628·13-s + 1.36·14-s + 0.295·15-s + 1.38·16-s − 0.229·17-s + 0.589·18-s − 0.121·19-s + 1.08·20-s + 0.445·21-s + 1.85·22-s + 1.73·23-s + 1.14·24-s − 0.738·25-s − 1.11·26-s + 0.192·27-s + 1.63·28-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

\(L(6)\) \(\approx\) \(11.88143315\)
\(L(\frac12)\) \(\approx\) \(11.88143315\)
\(L(\frac{13}{2})\) 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 \( 1 - 79.9T + 2.04e3T^{2} \)
5 \( 1 - 3.57e3T + 4.88e7T^{2} \)
7 \( 1 - 3.42e4T + 1.97e9T^{2} \)
11 \( 1 - 5.61e5T + 2.85e11T^{2} \)
13 \( 1 + 8.41e5T + 1.79e12T^{2} \)
17 \( 1 + 1.34e6T + 3.42e13T^{2} \)
19 \( 1 + 1.30e6T + 1.16e14T^{2} \)
23 \( 1 - 5.35e7T + 9.52e14T^{2} \)
29 \( 1 - 7.22e7T + 1.22e16T^{2} \)
31 \( 1 - 1.64e8T + 2.54e16T^{2} \)
37 \( 1 + 2.58e8T + 1.77e17T^{2} \)
41 \( 1 + 5.88e8T + 5.50e17T^{2} \)
43 \( 1 - 6.19e8T + 9.29e17T^{2} \)
47 \( 1 + 1.16e9T + 2.47e18T^{2} \)
53 \( 1 + 1.57e9T + 9.26e18T^{2} \)
61 \( 1 - 7.71e9T + 4.35e19T^{2} \)
67 \( 1 + 3.30e9T + 1.22e20T^{2} \)
71 \( 1 + 1.03e10T + 2.31e20T^{2} \)
73 \( 1 - 2.59e10T + 3.13e20T^{2} \)
79 \( 1 - 4.26e9T + 7.47e20T^{2} \)
83 \( 1 + 3.07e10T + 1.28e21T^{2} \)
89 \( 1 + 8.23e10T + 2.77e21T^{2} \)
97 \( 1 - 1.52e10T + 7.15e21T^{2} \)
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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

−11.09004492454401455748984385994, −9.787949785050480369159699889081, −8.550629241256922648866226361678, −7.18655241877490319504781700523, −6.37853118979307573678767138255, −5.13437051301394414077894507609, −4.42794151383462559496605975247, −3.29714342335399070872678460636, −2.29017387670643206785266718984, −1.34193984115932125397687450791, 1.34193984115932125397687450791, 2.29017387670643206785266718984, 3.29714342335399070872678460636, 4.42794151383462559496605975247, 5.13437051301394414077894507609, 6.37853118979307573678767138255, 7.18655241877490319504781700523, 8.550629241256922648866226361678, 9.787949785050480369159699889081, 11.09004492454401455748984385994

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