Properties

Label 2-177-1.1-c11-0-25
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  − 67.3·2-s + 243·3-s + 2.48e3·4-s + 9.96e3·5-s − 1.63e4·6-s − 3.76e4·7-s − 2.96e4·8-s + 5.90e4·9-s − 6.70e5·10-s + 1.81e5·11-s + 6.04e5·12-s − 1.08e4·13-s + 2.53e6·14-s + 2.42e6·15-s − 3.10e6·16-s − 8.94e6·17-s − 3.97e6·18-s − 4.83e5·19-s + 2.47e7·20-s − 9.13e6·21-s − 1.21e7·22-s − 5.76e7·23-s − 7.19e6·24-s + 5.03e7·25-s + 7.33e5·26-s + 1.43e7·27-s − 9.35e7·28-s + ⋯
L(s)  = 1  − 1.48·2-s + 0.577·3-s + 1.21·4-s + 1.42·5-s − 0.859·6-s − 0.845·7-s − 0.319·8-s + 0.333·9-s − 2.12·10-s + 0.338·11-s + 0.701·12-s − 0.00814·13-s + 1.25·14-s + 0.823·15-s − 0.739·16-s − 1.52·17-s − 0.496·18-s − 0.0448·19-s + 1.73·20-s − 0.488·21-s − 0.504·22-s − 1.86·23-s − 0.184·24-s + 1.03·25-s + 0.0121·26-s + 0.192·27-s − 1.02·28-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

\(L(6)\) \(\approx\) \(1.295055042\)
\(L(\frac12)\) \(\approx\) \(1.295055042\)
\(L(\frac{13}{2})\) 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 \( 1 + 67.3T + 2.04e3T^{2} \)
5 \( 1 - 9.96e3T + 4.88e7T^{2} \)
7 \( 1 + 3.76e4T + 1.97e9T^{2} \)
11 \( 1 - 1.81e5T + 2.85e11T^{2} \)
13 \( 1 + 1.08e4T + 1.79e12T^{2} \)
17 \( 1 + 8.94e6T + 3.42e13T^{2} \)
19 \( 1 + 4.83e5T + 1.16e14T^{2} \)
23 \( 1 + 5.76e7T + 9.52e14T^{2} \)
29 \( 1 + 7.44e7T + 1.22e16T^{2} \)
31 \( 1 + 3.78e6T + 2.54e16T^{2} \)
37 \( 1 - 2.66e8T + 1.77e17T^{2} \)
41 \( 1 - 9.21e8T + 5.50e17T^{2} \)
43 \( 1 - 1.72e9T + 9.29e17T^{2} \)
47 \( 1 - 5.56e8T + 2.47e18T^{2} \)
53 \( 1 + 5.10e8T + 9.26e18T^{2} \)
61 \( 1 - 4.70e9T + 4.35e19T^{2} \)
67 \( 1 - 6.76e8T + 1.22e20T^{2} \)
71 \( 1 - 2.31e10T + 2.31e20T^{2} \)
73 \( 1 - 1.24e10T + 3.13e20T^{2} \)
79 \( 1 + 1.81e10T + 7.47e20T^{2} \)
83 \( 1 - 2.88e10T + 1.28e21T^{2} \)
89 \( 1 - 1.17e10T + 2.77e21T^{2} \)
97 \( 1 + 6.35e9T + 7.15e21T^{2} \)
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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

−10.17485373810900985804051024037, −9.452971028647746630319075779664, −9.072576333265369807093485975375, −7.904031502182044825466675984154, −6.72755604762865629183494954450, −5.97156811471960568582692938840, −4.14903836193596546903265628676, −2.42660978015636713701694084657, −1.92040682291914841426254026659, −0.61308598049960024518813939729, 0.61308598049960024518813939729, 1.92040682291914841426254026659, 2.42660978015636713701694084657, 4.14903836193596546903265628676, 5.97156811471960568582692938840, 6.72755604762865629183494954450, 7.904031502182044825466675984154, 9.072576333265369807093485975375, 9.452971028647746630319075779664, 10.17485373810900985804051024037

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