Properties

Label 2-177-1.1-c11-0-88
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  − 17.2·2-s + 243·3-s − 1.75e3·4-s + 7.99e3·5-s − 4.19e3·6-s + 1.32e3·7-s + 6.55e4·8-s + 5.90e4·9-s − 1.37e5·10-s + 1.34e5·11-s − 4.25e5·12-s − 2.98e4·13-s − 2.28e4·14-s + 1.94e6·15-s + 2.45e6·16-s − 3.25e6·17-s − 1.01e6·18-s − 1.09e7·19-s − 1.39e7·20-s + 3.22e5·21-s − 2.32e6·22-s + 1.85e5·23-s + 1.59e7·24-s + 1.50e7·25-s + 5.14e5·26-s + 1.43e7·27-s − 2.32e6·28-s + ⋯
L(s)  = 1  − 0.381·2-s + 0.577·3-s − 0.854·4-s + 1.14·5-s − 0.219·6-s + 0.0298·7-s + 0.706·8-s + 0.333·9-s − 0.435·10-s + 0.252·11-s − 0.493·12-s − 0.0222·13-s − 0.0113·14-s + 0.660·15-s + 0.585·16-s − 0.556·17-s − 0.127·18-s − 1.01·19-s − 0.978·20-s + 0.0172·21-s − 0.0962·22-s + 0.00599·23-s + 0.408·24-s + 0.309·25-s + 0.00849·26-s + 0.192·27-s − 0.0254·28-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 \( 1 + 17.2T + 2.04e3T^{2} \)
5 \( 1 - 7.99e3T + 4.88e7T^{2} \)
7 \( 1 - 1.32e3T + 1.97e9T^{2} \)
11 \( 1 - 1.34e5T + 2.85e11T^{2} \)
13 \( 1 + 2.98e4T + 1.79e12T^{2} \)
17 \( 1 + 3.25e6T + 3.42e13T^{2} \)
19 \( 1 + 1.09e7T + 1.16e14T^{2} \)
23 \( 1 - 1.85e5T + 9.52e14T^{2} \)
29 \( 1 - 1.68e7T + 1.22e16T^{2} \)
31 \( 1 + 5.53e7T + 2.54e16T^{2} \)
37 \( 1 + 7.21e8T + 1.77e17T^{2} \)
41 \( 1 + 8.63e8T + 5.50e17T^{2} \)
43 \( 1 - 1.00e9T + 9.29e17T^{2} \)
47 \( 1 - 3.01e9T + 2.47e18T^{2} \)
53 \( 1 - 2.68e9T + 9.26e18T^{2} \)
61 \( 1 + 1.08e10T + 4.35e19T^{2} \)
67 \( 1 - 1.61e10T + 1.22e20T^{2} \)
71 \( 1 - 3.75e9T + 2.31e20T^{2} \)
73 \( 1 + 2.36e10T + 3.13e20T^{2} \)
79 \( 1 + 4.18e10T + 7.47e20T^{2} \)
83 \( 1 - 2.14e10T + 1.28e21T^{2} \)
89 \( 1 - 2.25e10T + 2.77e21T^{2} \)
97 \( 1 + 3.27e10T + 7.15e21T^{2} \)
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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

−9.992128187458102396393961777625, −9.055615707881975180610432129626, −8.558186477754223123654988340762, −7.23071945195429669664748811419, −6.00076262762479573141287043403, −4.85083498895938736745480963717, −3.76302727562429135875668769663, −2.28484511234791885545610896632, −1.38676654156033903246559704398, 0, 1.38676654156033903246559704398, 2.28484511234791885545610896632, 3.76302727562429135875668769663, 4.85083498895938736745480963717, 6.00076262762479573141287043403, 7.23071945195429669664748811419, 8.558186477754223123654988340762, 9.055615707881975180610432129626, 9.992128187458102396393961777625

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