Properties

Label 2-177-1.1-c11-0-0
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  + 52.4·2-s − 243·3-s + 700.·4-s − 9.19e3·5-s − 1.27e4·6-s − 2.46e4·7-s − 7.06e4·8-s + 5.90e4·9-s − 4.82e5·10-s + 3.49e5·11-s − 1.70e5·12-s − 1.19e6·13-s − 1.29e6·14-s + 2.23e6·15-s − 5.13e6·16-s − 1.08e7·17-s + 3.09e6·18-s − 3.01e4·19-s − 6.44e6·20-s + 5.99e6·21-s + 1.83e7·22-s − 5.27e7·23-s + 1.71e7·24-s + 3.58e7·25-s − 6.25e7·26-s − 1.43e7·27-s − 1.72e7·28-s + ⋯
L(s)  = 1  + 1.15·2-s − 0.577·3-s + 0.342·4-s − 1.31·5-s − 0.668·6-s − 0.554·7-s − 0.762·8-s + 0.333·9-s − 1.52·10-s + 0.653·11-s − 0.197·12-s − 0.891·13-s − 0.642·14-s + 0.760·15-s − 1.22·16-s − 1.85·17-s + 0.386·18-s − 0.00279·19-s − 0.450·20-s + 0.320·21-s + 0.757·22-s − 1.71·23-s + 0.439·24-s + 0.733·25-s − 1.03·26-s − 0.192·27-s − 0.189·28-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

\(L(6)\) \(\approx\) \(0.02467613477\)
\(L(\frac12)\) \(\approx\) \(0.02467613477\)
\(L(\frac{13}{2})\) 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 \( 1 - 52.4T + 2.04e3T^{2} \)
5 \( 1 + 9.19e3T + 4.88e7T^{2} \)
7 \( 1 + 2.46e4T + 1.97e9T^{2} \)
11 \( 1 - 3.49e5T + 2.85e11T^{2} \)
13 \( 1 + 1.19e6T + 1.79e12T^{2} \)
17 \( 1 + 1.08e7T + 3.42e13T^{2} \)
19 \( 1 + 3.01e4T + 1.16e14T^{2} \)
23 \( 1 + 5.27e7T + 9.52e14T^{2} \)
29 \( 1 + 1.41e8T + 1.22e16T^{2} \)
31 \( 1 + 1.52e8T + 2.54e16T^{2} \)
37 \( 1 - 3.15e7T + 1.77e17T^{2} \)
41 \( 1 + 1.01e9T + 5.50e17T^{2} \)
43 \( 1 + 2.79e8T + 9.29e17T^{2} \)
47 \( 1 - 1.89e9T + 2.47e18T^{2} \)
53 \( 1 - 2.66e9T + 9.26e18T^{2} \)
61 \( 1 - 9.53e9T + 4.35e19T^{2} \)
67 \( 1 - 1.58e10T + 1.22e20T^{2} \)
71 \( 1 + 3.96e9T + 2.31e20T^{2} \)
73 \( 1 - 1.73e10T + 3.13e20T^{2} \)
79 \( 1 + 4.62e10T + 7.47e20T^{2} \)
83 \( 1 - 8.12e9T + 1.28e21T^{2} \)
89 \( 1 + 4.10e10T + 2.77e21T^{2} \)
97 \( 1 + 5.13e10T + 7.15e21T^{2} \)
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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

−11.20516249164792740492243238599, −9.732929503128738867422534006165, −8.623973994784937206312107668007, −7.23189396988233489548604645261, −6.42371518719885517486661546938, −5.25510547158029866962153280670, −4.13196704888097982160948924967, −3.76422925103666985357429788717, −2.22394943108388739644682043332, −0.05725170861156419153174421816, 0.05725170861156419153174421816, 2.22394943108388739644682043332, 3.76422925103666985357429788717, 4.13196704888097982160948924967, 5.25510547158029866962153280670, 6.42371518719885517486661546938, 7.23189396988233489548604645261, 8.623973994784937206312107668007, 9.732929503128738867422534006165, 11.20516249164792740492243238599

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