Properties

Label 2-177-1.1-c13-0-116
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 101.·2-s + 729·3-s + 2.01e3·4-s + 2.20e4·5-s + 7.36e4·6-s + 3.32e5·7-s − 6.24e5·8-s + 5.31e5·9-s + 2.22e6·10-s − 8.68e6·11-s + 1.46e6·12-s + 9.77e6·13-s + 3.36e7·14-s + 1.60e7·15-s − 7.95e7·16-s + 6.60e6·17-s + 5.36e7·18-s − 1.51e8·19-s + 4.44e7·20-s + 2.42e8·21-s − 8.76e8·22-s − 4.72e8·23-s − 4.55e8·24-s − 7.33e8·25-s + 9.87e8·26-s + 3.87e8·27-s + 6.69e8·28-s + ⋯
L(s)  = 1  + 1.11·2-s + 0.577·3-s + 0.245·4-s + 0.631·5-s + 0.644·6-s + 1.06·7-s − 0.841·8-s + 0.333·9-s + 0.704·10-s − 1.47·11-s + 0.141·12-s + 0.561·13-s + 1.19·14-s + 0.364·15-s − 1.18·16-s + 0.0663·17-s + 0.372·18-s − 0.736·19-s + 0.155·20-s + 0.617·21-s − 1.64·22-s − 0.664·23-s − 0.486·24-s − 0.601·25-s + 0.626·26-s + 0.192·27-s + 0.262·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 - 4.21e10T \)
good2 \( 1 - 101.T + 8.19e3T^{2} \)
5 \( 1 - 2.20e4T + 1.22e9T^{2} \)
7 \( 1 - 3.32e5T + 9.68e10T^{2} \)
11 \( 1 + 8.68e6T + 3.45e13T^{2} \)
13 \( 1 - 9.77e6T + 3.02e14T^{2} \)
17 \( 1 - 6.60e6T + 9.90e15T^{2} \)
19 \( 1 + 1.51e8T + 4.20e16T^{2} \)
23 \( 1 + 4.72e8T + 5.04e17T^{2} \)
29 \( 1 - 4.96e9T + 1.02e19T^{2} \)
31 \( 1 + 2.82e9T + 2.44e19T^{2} \)
37 \( 1 + 2.05e10T + 2.43e20T^{2} \)
41 \( 1 + 3.02e10T + 9.25e20T^{2} \)
43 \( 1 + 5.85e10T + 1.71e21T^{2} \)
47 \( 1 - 7.37e10T + 5.46e21T^{2} \)
53 \( 1 + 1.58e11T + 2.60e22T^{2} \)
61 \( 1 + 5.95e11T + 1.61e23T^{2} \)
67 \( 1 - 8.35e11T + 5.48e23T^{2} \)
71 \( 1 - 9.96e11T + 1.16e24T^{2} \)
73 \( 1 - 2.49e12T + 1.67e24T^{2} \)
79 \( 1 + 2.96e12T + 4.66e24T^{2} \)
83 \( 1 - 1.16e12T + 8.87e24T^{2} \)
89 \( 1 + 6.46e12T + 2.19e25T^{2} \)
97 \( 1 - 7.75e12T + 6.73e25T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.955218150010129109304583835014, −8.625613141496940675618651482796, −8.008625144659048147622235522022, −6.51473265683535363623362179647, −5.40152718823386166308939608545, −4.76531830661235616315096459059, −3.62652104038847736555803908698, −2.52563064953331326591243985823, −1.68372569886506145768743007131, 0, 1.68372569886506145768743007131, 2.52563064953331326591243985823, 3.62652104038847736555803908698, 4.76531830661235616315096459059, 5.40152718823386166308939608545, 6.51473265683535363623362179647, 8.008625144659048147622235522022, 8.625613141496940675618651482796, 9.955218150010129109304583835014

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