Properties

Label 2-177-1.1-c11-0-11
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  + 18.0·2-s − 243·3-s − 1.72e3·4-s − 3.87e3·5-s − 4.37e3·6-s + 3.21e4·7-s − 6.79e4·8-s + 5.90e4·9-s − 6.99e4·10-s − 3.62e5·11-s + 4.18e5·12-s − 2.86e5·13-s + 5.78e5·14-s + 9.42e5·15-s + 2.30e6·16-s + 9.02e6·17-s + 1.06e6·18-s − 2.13e7·19-s + 6.68e6·20-s − 7.80e6·21-s − 6.53e6·22-s − 5.38e7·23-s + 1.65e7·24-s − 3.37e7·25-s − 5.15e6·26-s − 1.43e7·27-s − 5.53e7·28-s + ⋯
L(s)  = 1  + 0.398·2-s − 0.577·3-s − 0.841·4-s − 0.555·5-s − 0.229·6-s + 0.722·7-s − 0.733·8-s + 0.333·9-s − 0.221·10-s − 0.678·11-s + 0.485·12-s − 0.213·13-s + 0.287·14-s + 0.320·15-s + 0.549·16-s + 1.54·17-s + 0.132·18-s − 1.97·19-s + 0.467·20-s − 0.417·21-s − 0.270·22-s − 1.74·23-s + 0.423·24-s − 0.691·25-s − 0.0850·26-s − 0.192·27-s − 0.607·28-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

\(L(6)\) \(\approx\) \(0.5752139530\)
\(L(\frac12)\) \(\approx\) \(0.5752139530\)
\(L(\frac{13}{2})\) 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 \( 1 - 18.0T + 2.04e3T^{2} \)
5 \( 1 + 3.87e3T + 4.88e7T^{2} \)
7 \( 1 - 3.21e4T + 1.97e9T^{2} \)
11 \( 1 + 3.62e5T + 2.85e11T^{2} \)
13 \( 1 + 2.86e5T + 1.79e12T^{2} \)
17 \( 1 - 9.02e6T + 3.42e13T^{2} \)
19 \( 1 + 2.13e7T + 1.16e14T^{2} \)
23 \( 1 + 5.38e7T + 9.52e14T^{2} \)
29 \( 1 - 1.58e8T + 1.22e16T^{2} \)
31 \( 1 + 1.44e8T + 2.54e16T^{2} \)
37 \( 1 + 1.68e8T + 1.77e17T^{2} \)
41 \( 1 + 1.21e9T + 5.50e17T^{2} \)
43 \( 1 + 6.48e8T + 9.29e17T^{2} \)
47 \( 1 - 1.87e8T + 2.47e18T^{2} \)
53 \( 1 + 2.62e9T + 9.26e18T^{2} \)
61 \( 1 + 3.26e9T + 4.35e19T^{2} \)
67 \( 1 - 1.11e10T + 1.22e20T^{2} \)
71 \( 1 + 1.81e10T + 2.31e20T^{2} \)
73 \( 1 + 4.66e9T + 3.13e20T^{2} \)
79 \( 1 - 3.85e10T + 7.47e20T^{2} \)
83 \( 1 - 4.48e10T + 1.28e21T^{2} \)
89 \( 1 - 8.91e10T + 2.77e21T^{2} \)
97 \( 1 + 1.14e11T + 7.15e21T^{2} \)
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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

−10.62536011203534995329977568880, −9.892133426687273394130672646578, −8.361339336372514842387511474944, −7.88566220539495974313901114808, −6.28180236690412689389904387822, −5.26032613973471741766316675967, −4.46163687346093482783285757083, −3.49937964376422357827764273925, −1.82782535021431123555295531877, −0.32958503739156362304835166218, 0.32958503739156362304835166218, 1.82782535021431123555295531877, 3.49937964376422357827764273925, 4.46163687346093482783285757083, 5.26032613973471741766316675967, 6.28180236690412689389904387822, 7.88566220539495974313901114808, 8.361339336372514842387511474944, 9.892133426687273394130672646578, 10.62536011203534995329977568880

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