Properties

Label 2-177-1.1-c11-0-9
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  − 79.8·2-s + 243·3-s + 4.33e3·4-s − 5.37e3·5-s − 1.94e4·6-s − 5.86e4·7-s − 1.82e5·8-s + 5.90e4·9-s + 4.29e5·10-s − 1.26e5·11-s + 1.05e6·12-s + 1.76e5·13-s + 4.68e6·14-s − 1.30e6·15-s + 5.71e6·16-s + 5.10e6·17-s − 4.71e6·18-s − 2.83e6·19-s − 2.33e7·20-s − 1.42e7·21-s + 1.00e7·22-s − 2.26e7·23-s − 4.43e7·24-s − 1.99e7·25-s − 1.40e7·26-s + 1.43e7·27-s − 2.53e8·28-s + ⋯
L(s)  = 1  − 1.76·2-s + 0.577·3-s + 2.11·4-s − 0.769·5-s − 1.01·6-s − 1.31·7-s − 1.97·8-s + 0.333·9-s + 1.35·10-s − 0.236·11-s + 1.22·12-s + 0.131·13-s + 2.32·14-s − 0.444·15-s + 1.36·16-s + 0.871·17-s − 0.588·18-s − 0.263·19-s − 1.62·20-s − 0.760·21-s + 0.416·22-s − 0.733·23-s − 1.13·24-s − 0.407·25-s − 0.232·26-s + 0.192·27-s − 2.78·28-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

\(L(6)\) \(\approx\) \(0.2680572683\)
\(L(\frac12)\) \(\approx\) \(0.2680572683\)
\(L(\frac{13}{2})\) 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 \( 1 + 79.8T + 2.04e3T^{2} \)
5 \( 1 + 5.37e3T + 4.88e7T^{2} \)
7 \( 1 + 5.86e4T + 1.97e9T^{2} \)
11 \( 1 + 1.26e5T + 2.85e11T^{2} \)
13 \( 1 - 1.76e5T + 1.79e12T^{2} \)
17 \( 1 - 5.10e6T + 3.42e13T^{2} \)
19 \( 1 + 2.83e6T + 1.16e14T^{2} \)
23 \( 1 + 2.26e7T + 9.52e14T^{2} \)
29 \( 1 + 1.40e8T + 1.22e16T^{2} \)
31 \( 1 + 5.94e7T + 2.54e16T^{2} \)
37 \( 1 - 1.51e8T + 1.77e17T^{2} \)
41 \( 1 - 4.61e6T + 5.50e17T^{2} \)
43 \( 1 + 1.18e9T + 9.29e17T^{2} \)
47 \( 1 - 4.27e8T + 2.47e18T^{2} \)
53 \( 1 + 9.35e7T + 9.26e18T^{2} \)
61 \( 1 + 2.35e8T + 4.35e19T^{2} \)
67 \( 1 + 9.33e9T + 1.22e20T^{2} \)
71 \( 1 - 3.93e9T + 2.31e20T^{2} \)
73 \( 1 + 2.05e10T + 3.13e20T^{2} \)
79 \( 1 + 4.02e10T + 7.47e20T^{2} \)
83 \( 1 + 1.16e10T + 1.28e21T^{2} \)
89 \( 1 + 3.48e9T + 2.77e21T^{2} \)
97 \( 1 + 7.99e10T + 7.15e21T^{2} \)
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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

−10.18576065020305629987614611521, −9.644404280368033904965013340608, −8.698047559472472400750915073998, −7.81642524938906439325705366140, −7.13215742193993081693308788598, −5.98962373637241573016235020489, −3.81847852937073192145275496026, −2.86884301801928686434705260957, −1.63135123463964358113200490057, −0.29315631943638403789182674147, 0.29315631943638403789182674147, 1.63135123463964358113200490057, 2.86884301801928686434705260957, 3.81847852937073192145275496026, 5.98962373637241573016235020489, 7.13215742193993081693308788598, 7.81642524938906439325705366140, 8.698047559472472400750915073998, 9.644404280368033904965013340608, 10.18576065020305629987614611521

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