Properties

Label 2-177-1.1-c11-0-41
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.61·2-s − 243·3-s − 1.95e3·4-s − 8.97e3·5-s − 2.33e3·6-s + 1.93e3·7-s − 3.85e4·8-s + 5.90e4·9-s − 8.63e4·10-s − 9.08e5·11-s + 4.75e5·12-s − 1.01e6·13-s + 1.86e4·14-s + 2.18e6·15-s + 3.63e6·16-s + 1.33e6·17-s + 5.67e5·18-s + 1.42e7·19-s + 1.75e7·20-s − 4.70e5·21-s − 8.74e6·22-s + 7.93e6·23-s + 9.35e6·24-s + 3.17e7·25-s − 9.74e6·26-s − 1.43e7·27-s − 3.78e6·28-s + ⋯
L(s)  = 1  + 0.212·2-s − 0.577·3-s − 0.954·4-s − 1.28·5-s − 0.122·6-s + 0.0435·7-s − 0.415·8-s + 0.333·9-s − 0.273·10-s − 1.70·11-s + 0.551·12-s − 0.757·13-s + 0.00924·14-s + 0.741·15-s + 0.866·16-s + 0.228·17-s + 0.0708·18-s + 1.32·19-s + 1.22·20-s − 0.0251·21-s − 0.361·22-s + 0.256·23-s + 0.239·24-s + 0.650·25-s − 0.160·26-s − 0.192·27-s − 0.0415·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 - 9.61T + 2.04e3T^{2} \)
5 \( 1 + 8.97e3T + 4.88e7T^{2} \)
7 \( 1 - 1.93e3T + 1.97e9T^{2} \)
11 \( 1 + 9.08e5T + 2.85e11T^{2} \)
13 \( 1 + 1.01e6T + 1.79e12T^{2} \)
17 \( 1 - 1.33e6T + 3.42e13T^{2} \)
19 \( 1 - 1.42e7T + 1.16e14T^{2} \)
23 \( 1 - 7.93e6T + 9.52e14T^{2} \)
29 \( 1 - 6.25e6T + 1.22e16T^{2} \)
31 \( 1 + 1.62e8T + 2.54e16T^{2} \)
37 \( 1 - 1.71e8T + 1.77e17T^{2} \)
41 \( 1 - 5.36e8T + 5.50e17T^{2} \)
43 \( 1 - 1.25e9T + 9.29e17T^{2} \)
47 \( 1 - 1.92e9T + 2.47e18T^{2} \)
53 \( 1 - 3.11e9T + 9.26e18T^{2} \)
61 \( 1 + 8.48e9T + 4.35e19T^{2} \)
67 \( 1 - 4.77e9T + 1.22e20T^{2} \)
71 \( 1 - 5.10e8T + 2.31e20T^{2} \)
73 \( 1 - 1.18e10T + 3.13e20T^{2} \)
79 \( 1 + 2.99e10T + 7.47e20T^{2} \)
83 \( 1 - 5.44e10T + 1.28e21T^{2} \)
89 \( 1 + 2.89e10T + 2.77e21T^{2} \)
97 \( 1 - 1.91e10T + 7.15e21T^{2} \)
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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

−10.25759438391578381996549620818, −9.196180059520656002525607756374, −7.83783351517254583152686248080, −7.47300278978670083938851209450, −5.59546849694477344794891073114, −4.93802133712712767983677314863, −3.93902719957780919891017715210, −2.80904850714799141884066334082, −0.76601521420538472466556276588, 0, 0.76601521420538472466556276588, 2.80904850714799141884066334082, 3.93902719957780919891017715210, 4.93802133712712767983677314863, 5.59546849694477344794891073114, 7.47300278978670083938851209450, 7.83783351517254583152686248080, 9.196180059520656002525607756374, 10.25759438391578381996549620818

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