Properties

Label 2-177-1.1-c11-0-67
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  − 76.9·2-s + 243·3-s + 3.87e3·4-s + 1.56e3·5-s − 1.86e4·6-s + 5.11e4·7-s − 1.40e5·8-s + 5.90e4·9-s − 1.20e5·10-s + 9.13e5·11-s + 9.40e5·12-s + 3.28e4·13-s − 3.93e6·14-s + 3.80e5·15-s + 2.86e6·16-s + 6.52e6·17-s − 4.54e6·18-s + 2.05e7·19-s + 6.06e6·20-s + 1.24e7·21-s − 7.03e7·22-s − 4.51e6·23-s − 3.40e7·24-s − 4.63e7·25-s − 2.52e6·26-s + 1.43e7·27-s + 1.97e8·28-s + ⋯
L(s)  = 1  − 1.70·2-s + 0.577·3-s + 1.89·4-s + 0.224·5-s − 0.981·6-s + 1.14·7-s − 1.51·8-s + 0.333·9-s − 0.380·10-s + 1.71·11-s + 1.09·12-s + 0.0245·13-s − 1.95·14-s + 0.129·15-s + 0.682·16-s + 1.11·17-s − 0.566·18-s + 1.90·19-s + 0.423·20-s + 0.663·21-s − 2.90·22-s − 0.146·23-s − 0.873·24-s − 0.949·25-s − 0.0417·26-s + 0.192·27-s + 2.17·28-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

\(L(6)\) \(\approx\) \(2.245674796\)
\(L(\frac12)\) \(\approx\) \(2.245674796\)
\(L(\frac{13}{2})\) 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 \( 1 + 76.9T + 2.04e3T^{2} \)
5 \( 1 - 1.56e3T + 4.88e7T^{2} \)
7 \( 1 - 5.11e4T + 1.97e9T^{2} \)
11 \( 1 - 9.13e5T + 2.85e11T^{2} \)
13 \( 1 - 3.28e4T + 1.79e12T^{2} \)
17 \( 1 - 6.52e6T + 3.42e13T^{2} \)
19 \( 1 - 2.05e7T + 1.16e14T^{2} \)
23 \( 1 + 4.51e6T + 9.52e14T^{2} \)
29 \( 1 - 1.26e7T + 1.22e16T^{2} \)
31 \( 1 - 2.56e8T + 2.54e16T^{2} \)
37 \( 1 + 2.39e7T + 1.77e17T^{2} \)
41 \( 1 - 6.93e8T + 5.50e17T^{2} \)
43 \( 1 - 8.57e8T + 9.29e17T^{2} \)
47 \( 1 - 2.59e9T + 2.47e18T^{2} \)
53 \( 1 - 3.92e9T + 9.26e18T^{2} \)
61 \( 1 - 1.27e9T + 4.35e19T^{2} \)
67 \( 1 - 9.16e9T + 1.22e20T^{2} \)
71 \( 1 + 1.96e10T + 2.31e20T^{2} \)
73 \( 1 + 3.06e10T + 3.13e20T^{2} \)
79 \( 1 + 3.72e10T + 7.47e20T^{2} \)
83 \( 1 + 4.45e10T + 1.28e21T^{2} \)
89 \( 1 - 2.20e10T + 2.77e21T^{2} \)
97 \( 1 + 8.48e10T + 7.15e21T^{2} \)
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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

−10.23145552033671224703091485140, −9.519830077559394591727859308900, −8.757879778046604231983420008303, −7.82945633444600398385330003900, −7.18380030004597866400964745910, −5.79256093601594707492561996915, −4.11490703270431147745012693774, −2.63639162686666859163128104569, −1.29412832606479943872863082017, −1.10542384960922891195550043175, 1.10542384960922891195550043175, 1.29412832606479943872863082017, 2.63639162686666859163128104569, 4.11490703270431147745012693774, 5.79256093601594707492561996915, 7.18380030004597866400964745910, 7.82945633444600398385330003900, 8.757879778046604231983420008303, 9.519830077559394591727859308900, 10.23145552033671224703091485140

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