Properties

Label 2-177-1.1-c11-0-55
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  + 61.2·2-s + 243·3-s + 1.70e3·4-s − 4.72e3·5-s + 1.48e4·6-s + 6.05e4·7-s − 2.09e4·8-s + 5.90e4·9-s − 2.89e5·10-s + 4.73e5·11-s + 4.14e5·12-s + 6.10e5·13-s + 3.71e6·14-s − 1.14e6·15-s − 4.77e6·16-s + 1.14e7·17-s + 3.61e6·18-s − 7.38e6·19-s − 8.06e6·20-s + 1.47e7·21-s + 2.90e7·22-s − 2.70e7·23-s − 5.08e6·24-s − 2.64e7·25-s + 3.74e7·26-s + 1.43e7·27-s + 1.03e8·28-s + ⋯
L(s)  = 1  + 1.35·2-s + 0.577·3-s + 0.833·4-s − 0.676·5-s + 0.781·6-s + 1.36·7-s − 0.225·8-s + 0.333·9-s − 0.916·10-s + 0.886·11-s + 0.480·12-s + 0.456·13-s + 1.84·14-s − 0.390·15-s − 1.13·16-s + 1.96·17-s + 0.451·18-s − 0.684·19-s − 0.563·20-s + 0.786·21-s + 1.20·22-s − 0.875·23-s − 0.130·24-s − 0.541·25-s + 0.617·26-s + 0.192·27-s + 1.13·28-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

\(L(6)\) \(\approx\) \(6.870965376\)
\(L(\frac12)\) \(\approx\) \(6.870965376\)
\(L(\frac{13}{2})\) 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 \( 1 - 61.2T + 2.04e3T^{2} \)
5 \( 1 + 4.72e3T + 4.88e7T^{2} \)
7 \( 1 - 6.05e4T + 1.97e9T^{2} \)
11 \( 1 - 4.73e5T + 2.85e11T^{2} \)
13 \( 1 - 6.10e5T + 1.79e12T^{2} \)
17 \( 1 - 1.14e7T + 3.42e13T^{2} \)
19 \( 1 + 7.38e6T + 1.16e14T^{2} \)
23 \( 1 + 2.70e7T + 9.52e14T^{2} \)
29 \( 1 - 1.49e7T + 1.22e16T^{2} \)
31 \( 1 + 8.45e6T + 2.54e16T^{2} \)
37 \( 1 - 7.46e8T + 1.77e17T^{2} \)
41 \( 1 - 1.73e7T + 5.50e17T^{2} \)
43 \( 1 + 1.74e7T + 9.29e17T^{2} \)
47 \( 1 - 4.85e8T + 2.47e18T^{2} \)
53 \( 1 + 5.23e8T + 9.26e18T^{2} \)
61 \( 1 - 5.78e9T + 4.35e19T^{2} \)
67 \( 1 - 1.56e10T + 1.22e20T^{2} \)
71 \( 1 - 2.68e10T + 2.31e20T^{2} \)
73 \( 1 - 1.19e10T + 3.13e20T^{2} \)
79 \( 1 - 9.69e9T + 7.47e20T^{2} \)
83 \( 1 - 2.95e10T + 1.28e21T^{2} \)
89 \( 1 - 1.71e10T + 2.77e21T^{2} \)
97 \( 1 - 7.11e10T + 7.15e21T^{2} \)
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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

−11.14004812253067092061441982290, −9.648258398827365576310883002938, −8.341919675637943768939122243948, −7.71692159025639284564235667256, −6.28450002794154164364179599345, −5.19378597606246088327610609094, −4.10616616937089823370586416429, −3.62861401352614805153223636560, −2.21915019128663677464901716393, −1.00273697809551774265355573730, 1.00273697809551774265355573730, 2.21915019128663677464901716393, 3.62861401352614805153223636560, 4.10616616937089823370586416429, 5.19378597606246088327610609094, 6.28450002794154164364179599345, 7.71692159025639284564235667256, 8.341919675637943768939122243948, 9.648258398827365576310883002938, 11.14004812253067092061441982290

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