Properties

Label 2-177-1.1-c11-0-29
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  + 65.8·2-s + 243·3-s + 2.28e3·4-s − 9.01e3·5-s + 1.59e4·6-s − 6.05e4·7-s + 1.55e4·8-s + 5.90e4·9-s − 5.93e5·10-s − 2.03e5·11-s + 5.55e5·12-s + 1.02e6·13-s − 3.98e6·14-s − 2.19e6·15-s − 3.65e6·16-s − 7.79e5·17-s + 3.88e6·18-s + 1.22e7·19-s − 2.05e7·20-s − 1.47e7·21-s − 1.33e7·22-s + 1.69e7·23-s + 3.78e6·24-s + 3.24e7·25-s + 6.75e7·26-s + 1.43e7·27-s − 1.38e8·28-s + ⋯
L(s)  = 1  + 1.45·2-s + 0.577·3-s + 1.11·4-s − 1.28·5-s + 0.839·6-s − 1.36·7-s + 0.168·8-s + 0.333·9-s − 1.87·10-s − 0.380·11-s + 0.644·12-s + 0.766·13-s − 1.98·14-s − 0.744·15-s − 0.871·16-s − 0.133·17-s + 0.484·18-s + 1.13·19-s − 1.43·20-s − 0.786·21-s − 0.553·22-s + 0.548·23-s + 0.0970·24-s + 0.663·25-s + 1.11·26-s + 0.192·27-s − 1.51·28-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

\(L(6)\) \(\approx\) \(3.562274278\)
\(L(\frac12)\) \(\approx\) \(3.562274278\)
\(L(\frac{13}{2})\) 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 \( 1 - 65.8T + 2.04e3T^{2} \)
5 \( 1 + 9.01e3T + 4.88e7T^{2} \)
7 \( 1 + 6.05e4T + 1.97e9T^{2} \)
11 \( 1 + 2.03e5T + 2.85e11T^{2} \)
13 \( 1 - 1.02e6T + 1.79e12T^{2} \)
17 \( 1 + 7.79e5T + 3.42e13T^{2} \)
19 \( 1 - 1.22e7T + 1.16e14T^{2} \)
23 \( 1 - 1.69e7T + 9.52e14T^{2} \)
29 \( 1 - 6.11e7T + 1.22e16T^{2} \)
31 \( 1 + 1.67e7T + 2.54e16T^{2} \)
37 \( 1 + 6.00e7T + 1.77e17T^{2} \)
41 \( 1 - 1.81e8T + 5.50e17T^{2} \)
43 \( 1 - 4.15e8T + 9.29e17T^{2} \)
47 \( 1 + 8.28e7T + 2.47e18T^{2} \)
53 \( 1 - 2.67e9T + 9.26e18T^{2} \)
61 \( 1 - 9.04e9T + 4.35e19T^{2} \)
67 \( 1 - 5.14e9T + 1.22e20T^{2} \)
71 \( 1 - 2.17e10T + 2.31e20T^{2} \)
73 \( 1 - 1.30e10T + 3.13e20T^{2} \)
79 \( 1 - 9.94e9T + 7.47e20T^{2} \)
83 \( 1 + 1.98e10T + 1.28e21T^{2} \)
89 \( 1 - 8.04e9T + 2.77e21T^{2} \)
97 \( 1 + 8.30e10T + 7.15e21T^{2} \)
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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

−11.03508953808099486155696263456, −9.654036979945436544304936347193, −8.549408369225402243094322859610, −7.34538623443914100539311582012, −6.47126618265060009863047151247, −5.24386975234253853175626712197, −3.97530720984038686514956756498, −3.45483161273430973882112255885, −2.65933731554985311161502536034, −0.64446966702814512877776198841, 0.64446966702814512877776198841, 2.65933731554985311161502536034, 3.45483161273430973882112255885, 3.97530720984038686514956756498, 5.24386975234253853175626712197, 6.47126618265060009863047151247, 7.34538623443914100539311582012, 8.549408369225402243094322859610, 9.654036979945436544304936347193, 11.03508953808099486155696263456

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