Properties

Label 2-177-1.1-c9-0-12
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  − 9.19·2-s − 81·3-s − 427.·4-s − 1.96e3·5-s + 744.·6-s + 1.19e4·7-s + 8.63e3·8-s + 6.56e3·9-s + 1.80e4·10-s − 1.31e4·11-s + 3.46e4·12-s − 4.23e4·13-s − 1.09e5·14-s + 1.58e5·15-s + 1.39e5·16-s + 4.30e5·17-s − 6.03e4·18-s − 2.24e5·19-s + 8.38e5·20-s − 9.65e5·21-s + 1.20e5·22-s − 3.39e4·23-s − 6.99e5·24-s + 1.89e6·25-s + 3.89e5·26-s − 5.31e5·27-s − 5.09e6·28-s + ⋯
L(s)  = 1  − 0.406·2-s − 0.577·3-s − 0.834·4-s − 1.40·5-s + 0.234·6-s + 1.87·7-s + 0.745·8-s + 0.333·9-s + 0.570·10-s − 0.270·11-s + 0.482·12-s − 0.411·13-s − 0.762·14-s + 0.810·15-s + 0.532·16-s + 1.25·17-s − 0.135·18-s − 0.395·19-s + 1.17·20-s − 1.08·21-s + 0.109·22-s − 0.0252·23-s − 0.430·24-s + 0.970·25-s + 0.167·26-s − 0.192·27-s − 1.56·28-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

\(L(5)\) \(\approx\) \(0.7644679400\)
\(L(\frac12)\) \(\approx\) \(0.7644679400\)
\(L(\frac{11}{2})\) 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 \( 1 + 9.19T + 512T^{2} \)
5 \( 1 + 1.96e3T + 1.95e6T^{2} \)
7 \( 1 - 1.19e4T + 4.03e7T^{2} \)
11 \( 1 + 1.31e4T + 2.35e9T^{2} \)
13 \( 1 + 4.23e4T + 1.06e10T^{2} \)
17 \( 1 - 4.30e5T + 1.18e11T^{2} \)
19 \( 1 + 2.24e5T + 3.22e11T^{2} \)
23 \( 1 + 3.39e4T + 1.80e12T^{2} \)
29 \( 1 + 3.34e6T + 1.45e13T^{2} \)
31 \( 1 - 4.19e5T + 2.64e13T^{2} \)
37 \( 1 + 1.22e7T + 1.29e14T^{2} \)
41 \( 1 + 2.21e7T + 3.27e14T^{2} \)
43 \( 1 - 2.53e7T + 5.02e14T^{2} \)
47 \( 1 + 2.03e7T + 1.11e15T^{2} \)
53 \( 1 + 6.12e6T + 3.29e15T^{2} \)
61 \( 1 + 1.65e8T + 1.16e16T^{2} \)
67 \( 1 - 2.84e8T + 2.72e16T^{2} \)
71 \( 1 - 6.82e7T + 4.58e16T^{2} \)
73 \( 1 - 2.97e7T + 5.88e16T^{2} \)
79 \( 1 + 1.80e8T + 1.19e17T^{2} \)
83 \( 1 + 4.94e8T + 1.86e17T^{2} \)
89 \( 1 + 9.96e7T + 3.50e17T^{2} \)
97 \( 1 - 1.25e9T + 7.60e17T^{2} \)
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

−11.07648110909979306106411188816, −10.18716313996501228834716811326, −8.729949638853475102841597775263, −7.903955783764356967452224505308, −7.44577618492266069780383739244, −5.35189486736830326783081653350, −4.68424024392056515503890429325, −3.73795954075610386113771257599, −1.59150000288343342815906184547, −0.50669541643851450821591996781, 0.50669541643851450821591996781, 1.59150000288343342815906184547, 3.73795954075610386113771257599, 4.68424024392056515503890429325, 5.35189486736830326783081653350, 7.44577618492266069780383739244, 7.903955783764356967452224505308, 8.729949638853475102841597775263, 10.18716313996501228834716811326, 11.07648110909979306106411188816

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