Properties

Label 2-177-1.1-c11-0-3
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  − 70.9·2-s − 243·3-s + 2.98e3·4-s + 3.03e3·5-s + 1.72e4·6-s − 6.20e4·7-s − 6.67e4·8-s + 5.90e4·9-s − 2.15e5·10-s + 2.99e5·11-s − 7.26e5·12-s − 1.56e6·13-s + 4.40e6·14-s − 7.36e5·15-s − 1.38e6·16-s − 5.69e5·17-s − 4.19e6·18-s − 2.09e7·19-s + 9.06e6·20-s + 1.50e7·21-s − 2.12e7·22-s + 2.52e7·23-s + 1.62e7·24-s − 3.96e7·25-s + 1.10e8·26-s − 1.43e7·27-s − 1.85e8·28-s + ⋯
L(s)  = 1  − 1.56·2-s − 0.577·3-s + 1.45·4-s + 0.433·5-s + 0.905·6-s − 1.39·7-s − 0.719·8-s + 0.333·9-s − 0.680·10-s + 0.560·11-s − 0.842·12-s − 1.16·13-s + 2.18·14-s − 0.250·15-s − 0.330·16-s − 0.0972·17-s − 0.522·18-s − 1.93·19-s + 0.633·20-s + 0.805·21-s − 0.878·22-s + 0.816·23-s + 0.415·24-s − 0.811·25-s + 1.83·26-s − 0.192·27-s − 2.03·28-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

\(L(6)\) \(\approx\) \(0.06598451389\)
\(L(\frac12)\) \(\approx\) \(0.06598451389\)
\(L(\frac{13}{2})\) 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 \( 1 + 70.9T + 2.04e3T^{2} \)
5 \( 1 - 3.03e3T + 4.88e7T^{2} \)
7 \( 1 + 6.20e4T + 1.97e9T^{2} \)
11 \( 1 - 2.99e5T + 2.85e11T^{2} \)
13 \( 1 + 1.56e6T + 1.79e12T^{2} \)
17 \( 1 + 5.69e5T + 3.42e13T^{2} \)
19 \( 1 + 2.09e7T + 1.16e14T^{2} \)
23 \( 1 - 2.52e7T + 9.52e14T^{2} \)
29 \( 1 + 8.20e7T + 1.22e16T^{2} \)
31 \( 1 + 1.20e8T + 2.54e16T^{2} \)
37 \( 1 - 3.75e8T + 1.77e17T^{2} \)
41 \( 1 - 3.35e8T + 5.50e17T^{2} \)
43 \( 1 - 1.05e9T + 9.29e17T^{2} \)
47 \( 1 + 5.28e8T + 2.47e18T^{2} \)
53 \( 1 + 5.65e9T + 9.26e18T^{2} \)
61 \( 1 + 7.64e9T + 4.35e19T^{2} \)
67 \( 1 + 9.27e9T + 1.22e20T^{2} \)
71 \( 1 + 1.76e10T + 2.31e20T^{2} \)
73 \( 1 - 8.95e8T + 3.13e20T^{2} \)
79 \( 1 - 1.22e9T + 7.47e20T^{2} \)
83 \( 1 + 6.96e10T + 1.28e21T^{2} \)
89 \( 1 - 1.97e10T + 2.77e21T^{2} \)
97 \( 1 + 1.55e11T + 7.15e21T^{2} \)
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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

−10.39214747234498225734688684648, −9.529710953172202825167898312236, −9.100795325891146137156163350708, −7.63393081424028073159492078870, −6.71655135890073388623539583264, −5.99968281887459369458754599822, −4.31673720439375572974514812517, −2.64440035966447355871739673200, −1.57084116695392349369424803890, −0.15099002271638858402024158444, 0.15099002271638858402024158444, 1.57084116695392349369424803890, 2.64440035966447355871739673200, 4.31673720439375572974514812517, 5.99968281887459369458754599822, 6.71655135890073388623539583264, 7.63393081424028073159492078870, 9.100795325891146137156163350708, 9.529710953172202825167898312236, 10.39214747234498225734688684648

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