Properties

Label 2-177-1.1-c11-0-14
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  − 41.9·2-s − 243·3-s − 287.·4-s − 4.79e3·5-s + 1.01e4·6-s − 4.09e4·7-s + 9.79e4·8-s + 5.90e4·9-s + 2.01e5·10-s + 2.61e5·11-s + 6.99e4·12-s − 3.20e5·13-s + 1.71e6·14-s + 1.16e6·15-s − 3.52e6·16-s + 8.00e6·17-s − 2.47e6·18-s + 4.69e6·19-s + 1.38e6·20-s + 9.95e6·21-s − 1.09e7·22-s + 2.01e7·23-s − 2.38e7·24-s − 2.57e7·25-s + 1.34e7·26-s − 1.43e7·27-s + 1.17e7·28-s + ⋯
L(s)  = 1  − 0.927·2-s − 0.577·3-s − 0.140·4-s − 0.686·5-s + 0.535·6-s − 0.921·7-s + 1.05·8-s + 0.333·9-s + 0.636·10-s + 0.490·11-s + 0.0811·12-s − 0.239·13-s + 0.854·14-s + 0.396·15-s − 0.839·16-s + 1.36·17-s − 0.309·18-s + 0.434·19-s + 0.0964·20-s + 0.531·21-s − 0.454·22-s + 0.651·23-s − 0.610·24-s − 0.528·25-s + 0.221·26-s − 0.192·27-s + 0.129·28-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

\(L(6)\) \(\approx\) \(0.5286463992\)
\(L(\frac12)\) \(\approx\) \(0.5286463992\)
\(L(\frac{13}{2})\) 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 \( 1 + 41.9T + 2.04e3T^{2} \)
5 \( 1 + 4.79e3T + 4.88e7T^{2} \)
7 \( 1 + 4.09e4T + 1.97e9T^{2} \)
11 \( 1 - 2.61e5T + 2.85e11T^{2} \)
13 \( 1 + 3.20e5T + 1.79e12T^{2} \)
17 \( 1 - 8.00e6T + 3.42e13T^{2} \)
19 \( 1 - 4.69e6T + 1.16e14T^{2} \)
23 \( 1 - 2.01e7T + 9.52e14T^{2} \)
29 \( 1 - 9.30e7T + 1.22e16T^{2} \)
31 \( 1 - 4.66e7T + 2.54e16T^{2} \)
37 \( 1 + 4.29e8T + 1.77e17T^{2} \)
41 \( 1 - 1.99e7T + 5.50e17T^{2} \)
43 \( 1 - 1.68e8T + 9.29e17T^{2} \)
47 \( 1 + 5.76e8T + 2.47e18T^{2} \)
53 \( 1 - 4.39e9T + 9.26e18T^{2} \)
61 \( 1 + 2.27e9T + 4.35e19T^{2} \)
67 \( 1 - 1.03e10T + 1.22e20T^{2} \)
71 \( 1 + 2.02e10T + 2.31e20T^{2} \)
73 \( 1 + 3.07e10T + 3.13e20T^{2} \)
79 \( 1 + 2.51e10T + 7.47e20T^{2} \)
83 \( 1 + 1.75e10T + 1.28e21T^{2} \)
89 \( 1 + 1.52e10T + 2.77e21T^{2} \)
97 \( 1 - 1.56e10T + 7.15e21T^{2} \)
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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

−10.33938660547965579499276278243, −9.772181634886022371169034596445, −8.749289977306908856320004007183, −7.66867371357843276385091320350, −6.85446407209584277728280889603, −5.52414294919680455447192983871, −4.29429258232598763568506148496, −3.20550055115224699667410019946, −1.33291719416143670022976926804, −0.44304840143787358746357002310, 0.44304840143787358746357002310, 1.33291719416143670022976926804, 3.20550055115224699667410019946, 4.29429258232598763568506148496, 5.52414294919680455447192983871, 6.85446407209584277728280889603, 7.66867371357843276385091320350, 8.749289977306908856320004007183, 9.772181634886022371169034596445, 10.33938660547965579499276278243

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