Properties

Label 2-177-1.1-c11-0-22
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  − 64.1·2-s − 243·3-s + 2.07e3·4-s − 6.30e3·5-s + 1.55e4·6-s + 8.46e4·7-s − 1.56e3·8-s + 5.90e4·9-s + 4.04e5·10-s − 9.02e5·11-s − 5.03e5·12-s − 1.37e6·13-s − 5.43e6·14-s + 1.53e6·15-s − 4.14e6·16-s + 6.10e6·17-s − 3.79e6·18-s + 1.53e7·19-s − 1.30e7·20-s − 2.05e7·21-s + 5.79e7·22-s + 5.45e7·23-s + 3.79e5·24-s − 9.04e6·25-s + 8.82e7·26-s − 1.43e7·27-s + 1.75e8·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 1.01·4-s − 0.902·5-s + 0.818·6-s + 1.90·7-s − 0.0168·8-s + 0.333·9-s + 1.28·10-s − 1.69·11-s − 0.584·12-s − 1.02·13-s − 2.69·14-s + 0.521·15-s − 0.987·16-s + 1.04·17-s − 0.472·18-s + 1.42·19-s − 0.913·20-s − 1.09·21-s + 2.39·22-s + 1.76·23-s + 0.00972·24-s − 0.185·25-s + 1.45·26-s − 0.192·27-s + 1.92·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.7530534544\)
\(L(\frac12)\) \(\approx\) \(0.7530534544\)
\(L(\frac{13}{2})\) 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 \( 1 + 64.1T + 2.04e3T^{2} \)
5 \( 1 + 6.30e3T + 4.88e7T^{2} \)
7 \( 1 - 8.46e4T + 1.97e9T^{2} \)
11 \( 1 + 9.02e5T + 2.85e11T^{2} \)
13 \( 1 + 1.37e6T + 1.79e12T^{2} \)
17 \( 1 - 6.10e6T + 3.42e13T^{2} \)
19 \( 1 - 1.53e7T + 1.16e14T^{2} \)
23 \( 1 - 5.45e7T + 9.52e14T^{2} \)
29 \( 1 - 3.20e7T + 1.22e16T^{2} \)
31 \( 1 - 2.01e8T + 2.54e16T^{2} \)
37 \( 1 - 6.20e8T + 1.77e17T^{2} \)
41 \( 1 + 1.18e9T + 5.50e17T^{2} \)
43 \( 1 + 2.91e8T + 9.29e17T^{2} \)
47 \( 1 - 6.14e7T + 2.47e18T^{2} \)
53 \( 1 + 3.65e9T + 9.26e18T^{2} \)
61 \( 1 - 1.13e10T + 4.35e19T^{2} \)
67 \( 1 + 8.11e9T + 1.22e20T^{2} \)
71 \( 1 + 1.75e10T + 2.31e20T^{2} \)
73 \( 1 + 1.78e10T + 3.13e20T^{2} \)
79 \( 1 - 2.32e10T + 7.47e20T^{2} \)
83 \( 1 - 8.67e9T + 1.28e21T^{2} \)
89 \( 1 - 7.88e10T + 2.77e21T^{2} \)
97 \( 1 + 6.50e10T + 7.15e21T^{2} \)
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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

−10.60033212285896145576687428489, −9.802390545496031336247741899772, −8.352736205860585406943474666810, −7.72727084770310313835266508767, −7.35277688667361182567706397497, −5.16618095301261905976851265131, −4.76864622466574981561009029470, −2.73634732121980471693027839923, −1.34712815620250782231852926704, −0.56932938228978206196610802612, 0.56932938228978206196610802612, 1.34712815620250782231852926704, 2.73634732121980471693027839923, 4.76864622466574981561009029470, 5.16618095301261905976851265131, 7.35277688667361182567706397497, 7.72727084770310313835266508767, 8.352736205860585406943474666810, 9.802390545496031336247741899772, 10.60033212285896145576687428489

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