Properties

Label 2-177-1.1-c11-0-34
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  − 7.28·2-s + 243·3-s − 1.99e3·4-s + 6.13e3·5-s − 1.77e3·6-s − 3.37e4·7-s + 2.94e4·8-s + 5.90e4·9-s − 4.47e4·10-s + 9.84e5·11-s − 4.84e5·12-s − 2.05e6·13-s + 2.45e5·14-s + 1.49e6·15-s + 3.87e6·16-s + 6.23e6·17-s − 4.30e5·18-s + 1.75e5·19-s − 1.22e7·20-s − 8.19e6·21-s − 7.16e6·22-s + 8.59e6·23-s + 7.15e6·24-s − 1.11e7·25-s + 1.49e7·26-s + 1.43e7·27-s + 6.72e7·28-s + ⋯
L(s)  = 1  − 0.160·2-s + 0.577·3-s − 0.974·4-s + 0.878·5-s − 0.0929·6-s − 0.758·7-s + 0.317·8-s + 0.333·9-s − 0.141·10-s + 1.84·11-s − 0.562·12-s − 1.53·13-s + 0.122·14-s + 0.507·15-s + 0.922·16-s + 1.06·17-s − 0.0536·18-s + 0.0162·19-s − 0.855·20-s − 0.437·21-s − 0.296·22-s + 0.278·23-s + 0.183·24-s − 0.228·25-s + 0.247·26-s + 0.192·27-s + 0.738·28-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

\(L(6)\) \(\approx\) \(2.326247016\)
\(L(\frac12)\) \(\approx\) \(2.326247016\)
\(L(\frac{13}{2})\) 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 \( 1 + 7.28T + 2.04e3T^{2} \)
5 \( 1 - 6.13e3T + 4.88e7T^{2} \)
7 \( 1 + 3.37e4T + 1.97e9T^{2} \)
11 \( 1 - 9.84e5T + 2.85e11T^{2} \)
13 \( 1 + 2.05e6T + 1.79e12T^{2} \)
17 \( 1 - 6.23e6T + 3.42e13T^{2} \)
19 \( 1 - 1.75e5T + 1.16e14T^{2} \)
23 \( 1 - 8.59e6T + 9.52e14T^{2} \)
29 \( 1 + 9.86e7T + 1.22e16T^{2} \)
31 \( 1 - 3.66e7T + 2.54e16T^{2} \)
37 \( 1 - 5.41e8T + 1.77e17T^{2} \)
41 \( 1 + 2.76e8T + 5.50e17T^{2} \)
43 \( 1 + 4.46e8T + 9.29e17T^{2} \)
47 \( 1 - 1.06e8T + 2.47e18T^{2} \)
53 \( 1 + 2.52e8T + 9.26e18T^{2} \)
61 \( 1 - 1.18e10T + 4.35e19T^{2} \)
67 \( 1 + 3.55e9T + 1.22e20T^{2} \)
71 \( 1 + 3.31e9T + 2.31e20T^{2} \)
73 \( 1 + 8.37e9T + 3.13e20T^{2} \)
79 \( 1 + 2.44e9T + 7.47e20T^{2} \)
83 \( 1 - 5.29e10T + 1.28e21T^{2} \)
89 \( 1 - 5.01e10T + 2.77e21T^{2} \)
97 \( 1 + 9.10e10T + 7.15e21T^{2} \)
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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

−9.991618652494012015228475819209, −9.638875943079083803298775560665, −9.034352600293649632820715671644, −7.73024561102436756431201612571, −6.58784686127125555368080665859, −5.43510890306800505898542542015, −4.20458840735147325022865677921, −3.21031747870351670142630509931, −1.83044248964636425698583079419, −0.71252571434290133016463775841, 0.71252571434290133016463775841, 1.83044248964636425698583079419, 3.21031747870351670142630509931, 4.20458840735147325022865677921, 5.43510890306800505898542542015, 6.58784686127125555368080665859, 7.73024561102436756431201612571, 9.034352600293649632820715671644, 9.638875943079083803298775560665, 9.991618652494012015228475819209

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