Properties

Label 2-177-1.1-c9-0-15
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 32.9·2-s + 81·3-s + 575.·4-s − 958.·5-s − 2.67e3·6-s − 5.51e3·7-s − 2.10e3·8-s + 6.56e3·9-s + 3.16e4·10-s + 2.93e4·11-s + 4.66e4·12-s + 1.37e5·13-s + 1.81e5·14-s − 7.76e4·15-s − 2.25e5·16-s − 3.39e5·17-s − 2.16e5·18-s + 8.62e4·19-s − 5.51e5·20-s − 4.46e5·21-s − 9.66e5·22-s + 1.13e6·23-s − 1.70e5·24-s − 1.03e6·25-s − 4.55e6·26-s + 5.31e5·27-s − 3.17e6·28-s + ⋯
L(s)  = 1  − 1.45·2-s + 0.577·3-s + 1.12·4-s − 0.685·5-s − 0.841·6-s − 0.867·7-s − 0.181·8-s + 0.333·9-s + 0.999·10-s + 0.603·11-s + 0.649·12-s + 1.33·13-s + 1.26·14-s − 0.395·15-s − 0.860·16-s − 0.986·17-s − 0.485·18-s + 0.151·19-s − 0.771·20-s − 0.501·21-s − 0.880·22-s + 0.845·23-s − 0.104·24-s − 0.529·25-s − 1.95·26-s + 0.192·27-s − 0.975·28-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

\(L(5)\) \(\approx\) \(0.8279116790\)
\(L(\frac12)\) \(\approx\) \(0.8279116790\)
\(L(\frac{11}{2})\) 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 \( 1 + 32.9T + 512T^{2} \)
5 \( 1 + 958.T + 1.95e6T^{2} \)
7 \( 1 + 5.51e3T + 4.03e7T^{2} \)
11 \( 1 - 2.93e4T + 2.35e9T^{2} \)
13 \( 1 - 1.37e5T + 1.06e10T^{2} \)
17 \( 1 + 3.39e5T + 1.18e11T^{2} \)
19 \( 1 - 8.62e4T + 3.22e11T^{2} \)
23 \( 1 - 1.13e6T + 1.80e12T^{2} \)
29 \( 1 + 9.19e5T + 1.45e13T^{2} \)
31 \( 1 - 1.08e6T + 2.64e13T^{2} \)
37 \( 1 + 2.63e6T + 1.29e14T^{2} \)
41 \( 1 - 9.33e6T + 3.27e14T^{2} \)
43 \( 1 - 6.15e6T + 5.02e14T^{2} \)
47 \( 1 + 5.67e7T + 1.11e15T^{2} \)
53 \( 1 - 5.09e7T + 3.29e15T^{2} \)
61 \( 1 + 9.11e7T + 1.16e16T^{2} \)
67 \( 1 + 9.97e7T + 2.72e16T^{2} \)
71 \( 1 - 1.19e8T + 4.58e16T^{2} \)
73 \( 1 + 6.25e6T + 5.88e16T^{2} \)
79 \( 1 + 2.70e8T + 1.19e17T^{2} \)
83 \( 1 + 4.06e8T + 1.86e17T^{2} \)
89 \( 1 - 9.88e8T + 3.50e17T^{2} \)
97 \( 1 + 1.09e9T + 7.60e17T^{2} \)
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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

−10.82529934224543496254633241232, −9.691297790823039023323605853666, −8.956805834561907865164004150754, −8.276121756272080035225111948369, −7.20243923875020173426246894174, −6.33979705365806666746617346564, −4.25199741511256758931590161473, −3.18130669348122895470669012603, −1.68910134600434932410759260745, −0.55628994279862697143417450978, 0.55628994279862697143417450978, 1.68910134600434932410759260745, 3.18130669348122895470669012603, 4.25199741511256758931590161473, 6.33979705365806666746617346564, 7.20243923875020173426246894174, 8.276121756272080035225111948369, 8.956805834561907865164004150754, 9.691297790823039023323605853666, 10.82529934224543496254633241232

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