Properties

Label 2-177-1.1-c11-0-10
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  − 5.14·2-s − 243·3-s − 2.02e3·4-s + 3.78e3·5-s + 1.24e3·6-s − 3.63e4·7-s + 2.09e4·8-s + 5.90e4·9-s − 1.94e4·10-s − 6.71e5·11-s + 4.91e5·12-s + 1.37e6·13-s + 1.86e5·14-s − 9.20e5·15-s + 4.03e6·16-s − 4.33e6·17-s − 3.03e5·18-s + 8.25e6·19-s − 7.66e6·20-s + 8.82e6·21-s + 3.45e6·22-s − 6.13e7·23-s − 5.08e6·24-s − 3.44e7·25-s − 7.07e6·26-s − 1.43e7·27-s + 7.33e7·28-s + ⋯
L(s)  = 1  − 0.113·2-s − 0.577·3-s − 0.987·4-s + 0.542·5-s + 0.0655·6-s − 0.816·7-s + 0.225·8-s + 0.333·9-s − 0.0616·10-s − 1.25·11-s + 0.569·12-s + 1.02·13-s + 0.0927·14-s − 0.313·15-s + 0.961·16-s − 0.739·17-s − 0.0378·18-s + 0.764·19-s − 0.535·20-s + 0.471·21-s + 0.142·22-s − 1.98·23-s − 0.130·24-s − 0.705·25-s − 0.116·26-s − 0.192·27-s + 0.805·28-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

\(L(6)\) \(\approx\) \(0.4202515602\)
\(L(\frac12)\) \(\approx\) \(0.4202515602\)
\(L(\frac{13}{2})\) 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 \( 1 + 5.14T + 2.04e3T^{2} \)
5 \( 1 - 3.78e3T + 4.88e7T^{2} \)
7 \( 1 + 3.63e4T + 1.97e9T^{2} \)
11 \( 1 + 6.71e5T + 2.85e11T^{2} \)
13 \( 1 - 1.37e6T + 1.79e12T^{2} \)
17 \( 1 + 4.33e6T + 3.42e13T^{2} \)
19 \( 1 - 8.25e6T + 1.16e14T^{2} \)
23 \( 1 + 6.13e7T + 9.52e14T^{2} \)
29 \( 1 + 7.51e7T + 1.22e16T^{2} \)
31 \( 1 - 2.73e8T + 2.54e16T^{2} \)
37 \( 1 + 5.71e8T + 1.77e17T^{2} \)
41 \( 1 + 9.81e8T + 5.50e17T^{2} \)
43 \( 1 - 1.36e9T + 9.29e17T^{2} \)
47 \( 1 + 1.13e9T + 2.47e18T^{2} \)
53 \( 1 + 3.65e9T + 9.26e18T^{2} \)
61 \( 1 - 4.73e9T + 4.35e19T^{2} \)
67 \( 1 + 1.18e10T + 1.22e20T^{2} \)
71 \( 1 + 2.10e10T + 2.31e20T^{2} \)
73 \( 1 + 4.85e9T + 3.13e20T^{2} \)
79 \( 1 + 3.22e10T + 7.47e20T^{2} \)
83 \( 1 + 3.61e10T + 1.28e21T^{2} \)
89 \( 1 + 1.60e10T + 2.77e21T^{2} \)
97 \( 1 - 1.28e11T + 7.15e21T^{2} \)
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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

−10.25966151743458995529923734467, −9.941325518146693684848527204711, −8.742963418107070680522400786141, −7.74365774326251734420338304966, −6.24670252913444613176073802405, −5.58197227660361192908641745350, −4.42987159640363063958460679389, −3.26364333921940516633246114534, −1.71919393330673666241683597197, −0.30705275236426650246694609403, 0.30705275236426650246694609403, 1.71919393330673666241683597197, 3.26364333921940516633246114534, 4.42987159640363063958460679389, 5.58197227660361192908641745350, 6.24670252913444613176073802405, 7.74365774326251734420338304966, 8.742963418107070680522400786141, 9.941325518146693684848527204711, 10.25966151743458995529923734467

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