Properties

Label 2-177-1.1-c11-0-39
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  − 41.5·2-s − 243·3-s − 318.·4-s + 9.65e3·5-s + 1.01e4·6-s − 1.15e4·7-s + 9.84e4·8-s + 5.90e4·9-s − 4.01e5·10-s − 5.21e5·11-s + 7.74e4·12-s + 2.19e6·13-s + 4.81e5·14-s − 2.34e6·15-s − 3.44e6·16-s − 5.78e6·17-s − 2.45e6·18-s + 1.37e7·19-s − 3.07e6·20-s + 2.81e6·21-s + 2.16e7·22-s + 5.21e7·23-s − 2.39e7·24-s + 4.44e7·25-s − 9.11e7·26-s − 1.43e7·27-s + 3.68e6·28-s + ⋯
L(s)  = 1  − 0.918·2-s − 0.577·3-s − 0.155·4-s + 1.38·5-s + 0.530·6-s − 0.260·7-s + 1.06·8-s + 0.333·9-s − 1.27·10-s − 0.975·11-s + 0.0898·12-s + 1.63·13-s + 0.239·14-s − 0.797·15-s − 0.820·16-s − 0.987·17-s − 0.306·18-s + 1.27·19-s − 0.214·20-s + 0.150·21-s + 0.896·22-s + 1.68·23-s − 0.613·24-s + 0.910·25-s − 1.50·26-s − 0.192·27-s + 0.0405·28-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

\(L(6)\) \(\approx\) \(1.448316240\)
\(L(\frac12)\) \(\approx\) \(1.448316240\)
\(L(\frac{13}{2})\) 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 \( 1 + 41.5T + 2.04e3T^{2} \)
5 \( 1 - 9.65e3T + 4.88e7T^{2} \)
7 \( 1 + 1.15e4T + 1.97e9T^{2} \)
11 \( 1 + 5.21e5T + 2.85e11T^{2} \)
13 \( 1 - 2.19e6T + 1.79e12T^{2} \)
17 \( 1 + 5.78e6T + 3.42e13T^{2} \)
19 \( 1 - 1.37e7T + 1.16e14T^{2} \)
23 \( 1 - 5.21e7T + 9.52e14T^{2} \)
29 \( 1 - 1.77e8T + 1.22e16T^{2} \)
31 \( 1 + 1.22e8T + 2.54e16T^{2} \)
37 \( 1 - 3.62e8T + 1.77e17T^{2} \)
41 \( 1 - 1.34e9T + 5.50e17T^{2} \)
43 \( 1 - 1.17e9T + 9.29e17T^{2} \)
47 \( 1 - 5.28e8T + 2.47e18T^{2} \)
53 \( 1 + 5.18e9T + 9.26e18T^{2} \)
61 \( 1 + 9.00e9T + 4.35e19T^{2} \)
67 \( 1 - 5.11e9T + 1.22e20T^{2} \)
71 \( 1 + 2.99e9T + 2.31e20T^{2} \)
73 \( 1 + 2.74e10T + 3.13e20T^{2} \)
79 \( 1 - 2.06e10T + 7.47e20T^{2} \)
83 \( 1 - 3.14e10T + 1.28e21T^{2} \)
89 \( 1 - 2.32e10T + 2.77e21T^{2} \)
97 \( 1 + 1.39e11T + 7.15e21T^{2} \)
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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

−10.67802745371422872550632550573, −9.508286507532419192306921186383, −9.023001390762507990333990125114, −7.75986319963808155169799072030, −6.51388412290240077151062621600, −5.60750254134643146281308154156, −4.59609532717405677905626759074, −2.83355099984316624429489561620, −1.43972999000294684773125356216, −0.72778073072673147384900847899, 0.72778073072673147384900847899, 1.43972999000294684773125356216, 2.83355099984316624429489561620, 4.59609532717405677905626759074, 5.60750254134643146281308154156, 6.51388412290240077151062621600, 7.75986319963808155169799072030, 9.023001390762507990333990125114, 9.508286507532419192306921186383, 10.67802745371422872550632550573

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