Properties

Label 2-177-1.1-c11-0-42
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  + 43.4·2-s − 243·3-s − 156.·4-s + 1.83e3·5-s − 1.05e4·6-s + 3.38e4·7-s − 9.58e4·8-s + 5.90e4·9-s + 7.97e4·10-s + 9.08e5·11-s + 3.81e4·12-s + 5.03e5·13-s + 1.47e6·14-s − 4.45e5·15-s − 3.84e6·16-s + 4.40e6·17-s + 2.56e6·18-s + 1.49e6·19-s − 2.87e5·20-s − 8.22e6·21-s + 3.94e7·22-s − 1.40e7·23-s + 2.33e7·24-s − 4.54e7·25-s + 2.18e7·26-s − 1.43e7·27-s − 5.31e6·28-s + ⋯
L(s)  = 1  + 0.960·2-s − 0.577·3-s − 0.0766·4-s + 0.262·5-s − 0.554·6-s + 0.761·7-s − 1.03·8-s + 0.333·9-s + 0.252·10-s + 1.70·11-s + 0.0442·12-s + 0.376·13-s + 0.731·14-s − 0.151·15-s − 0.917·16-s + 0.752·17-s + 0.320·18-s + 0.138·19-s − 0.0201·20-s − 0.439·21-s + 1.63·22-s − 0.455·23-s + 0.597·24-s − 0.931·25-s + 0.361·26-s − 0.192·27-s − 0.0583·28-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

\(L(6)\) \(\approx\) \(3.419474226\)
\(L(\frac12)\) \(\approx\) \(3.419474226\)
\(L(\frac{13}{2})\) 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 \( 1 - 43.4T + 2.04e3T^{2} \)
5 \( 1 - 1.83e3T + 4.88e7T^{2} \)
7 \( 1 - 3.38e4T + 1.97e9T^{2} \)
11 \( 1 - 9.08e5T + 2.85e11T^{2} \)
13 \( 1 - 5.03e5T + 1.79e12T^{2} \)
17 \( 1 - 4.40e6T + 3.42e13T^{2} \)
19 \( 1 - 1.49e6T + 1.16e14T^{2} \)
23 \( 1 + 1.40e7T + 9.52e14T^{2} \)
29 \( 1 + 7.57e7T + 1.22e16T^{2} \)
31 \( 1 + 1.41e8T + 2.54e16T^{2} \)
37 \( 1 + 4.72e8T + 1.77e17T^{2} \)
41 \( 1 - 1.06e9T + 5.50e17T^{2} \)
43 \( 1 - 1.00e9T + 9.29e17T^{2} \)
47 \( 1 - 1.98e9T + 2.47e18T^{2} \)
53 \( 1 - 4.29e9T + 9.26e18T^{2} \)
61 \( 1 - 6.72e8T + 4.35e19T^{2} \)
67 \( 1 + 5.16e9T + 1.22e20T^{2} \)
71 \( 1 + 1.10e10T + 2.31e20T^{2} \)
73 \( 1 - 1.07e10T + 3.13e20T^{2} \)
79 \( 1 - 5.11e10T + 7.47e20T^{2} \)
83 \( 1 + 6.20e10T + 1.28e21T^{2} \)
89 \( 1 - 4.27e10T + 2.77e21T^{2} \)
97 \( 1 - 1.93e10T + 7.15e21T^{2} \)
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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

−11.00661664668225732934010619782, −9.625445056971150776579579234192, −8.806840565782343628948452417088, −7.36938243520911524141018508899, −6.08237349527034369145874740892, −5.51575224484254580566684024840, −4.28283954633977940878200457456, −3.64128353811300947691458571809, −1.88654040157261111224040077174, −0.78066881283115041094195855392, 0.78066881283115041094195855392, 1.88654040157261111224040077174, 3.64128353811300947691458571809, 4.28283954633977940878200457456, 5.51575224484254580566684024840, 6.08237349527034369145874740892, 7.36938243520911524141018508899, 8.806840565782343628948452417088, 9.625445056971150776579579234192, 11.00661664668225732934010619782

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