Properties

Label 2-177-1.1-c9-0-21
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 7.89·2-s − 81·3-s − 449.·4-s + 876.·5-s + 639.·6-s − 1.20e3·7-s + 7.59e3·8-s + 6.56e3·9-s − 6.92e3·10-s + 3.88e4·11-s + 3.64e4·12-s + 1.83e5·13-s + 9.49e3·14-s − 7.09e4·15-s + 1.70e5·16-s + 1.39e5·17-s − 5.18e4·18-s − 8.67e5·19-s − 3.94e5·20-s + 9.73e4·21-s − 3.06e5·22-s + 1.16e6·23-s − 6.15e5·24-s − 1.18e6·25-s − 1.44e6·26-s − 5.31e5·27-s + 5.40e5·28-s + ⋯
L(s)  = 1  − 0.348·2-s − 0.577·3-s − 0.878·4-s + 0.627·5-s + 0.201·6-s − 0.189·7-s + 0.655·8-s + 0.333·9-s − 0.218·10-s + 0.799·11-s + 0.507·12-s + 1.78·13-s + 0.0660·14-s − 0.362·15-s + 0.649·16-s + 0.405·17-s − 0.116·18-s − 1.52·19-s − 0.550·20-s + 0.109·21-s − 0.278·22-s + 0.869·23-s − 0.378·24-s − 0.606·25-s − 0.621·26-s − 0.192·27-s + 0.166·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.463380594\)
\(L(\frac12)\) \(\approx\) \(1.463380594\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
59 \( 1 - 1.21e7T \)
good2 \( 1 + 7.89T + 512T^{2} \)
5 \( 1 - 876.T + 1.95e6T^{2} \)
7 \( 1 + 1.20e3T + 4.03e7T^{2} \)
11 \( 1 - 3.88e4T + 2.35e9T^{2} \)
13 \( 1 - 1.83e5T + 1.06e10T^{2} \)
17 \( 1 - 1.39e5T + 1.18e11T^{2} \)
19 \( 1 + 8.67e5T + 3.22e11T^{2} \)
23 \( 1 - 1.16e6T + 1.80e12T^{2} \)
29 \( 1 - 2.91e6T + 1.45e13T^{2} \)
31 \( 1 - 4.99e6T + 2.64e13T^{2} \)
37 \( 1 + 1.18e6T + 1.29e14T^{2} \)
41 \( 1 + 1.41e7T + 3.27e14T^{2} \)
43 \( 1 + 1.70e7T + 5.02e14T^{2} \)
47 \( 1 - 1.28e7T + 1.11e15T^{2} \)
53 \( 1 - 7.27e7T + 3.29e15T^{2} \)
61 \( 1 + 9.05e7T + 1.16e16T^{2} \)
67 \( 1 + 1.29e7T + 2.72e16T^{2} \)
71 \( 1 - 2.92e8T + 4.58e16T^{2} \)
73 \( 1 + 4.07e8T + 5.88e16T^{2} \)
79 \( 1 - 2.64e8T + 1.19e17T^{2} \)
83 \( 1 + 2.68e8T + 1.86e17T^{2} \)
89 \( 1 - 6.83e8T + 3.50e17T^{2} \)
97 \( 1 - 5.92e8T + 7.60e17T^{2} \)
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

−10.78241768522433898036051660312, −10.04354303182917085463377591419, −8.983434175726170635087222979404, −8.294511395944761697319553954929, −6.62801009055504783207215745109, −5.89092251602079568231456769345, −4.61554331369367529824318416437, −3.58188983148670416638293689040, −1.60970473404038771787485653126, −0.70819950617079958604687574660, 0.70819950617079958604687574660, 1.60970473404038771787485653126, 3.58188983148670416638293689040, 4.61554331369367529824318416437, 5.89092251602079568231456769345, 6.62801009055504783207215745109, 8.294511395944761697319553954929, 8.983434175726170635087222979404, 10.04354303182917085463377591419, 10.78241768522433898036051660312

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