Properties

Label 2-177-1.1-c13-0-97
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  − 112.·2-s + 729·3-s + 4.54e3·4-s + 3.76e4·5-s − 8.22e4·6-s − 6.04e4·7-s + 4.11e5·8-s + 5.31e5·9-s − 4.24e6·10-s − 5.38e6·11-s + 3.31e6·12-s + 1.78e7·13-s + 6.82e6·14-s + 2.74e7·15-s − 8.36e7·16-s − 7.26e7·17-s − 5.99e7·18-s + 2.05e8·19-s + 1.71e8·20-s − 4.40e7·21-s + 6.07e8·22-s − 2.11e8·23-s + 2.99e8·24-s + 1.93e8·25-s − 2.01e9·26-s + 3.87e8·27-s − 2.75e8·28-s + ⋯
L(s)  = 1  − 1.24·2-s + 0.577·3-s + 0.555·4-s + 1.07·5-s − 0.720·6-s − 0.194·7-s + 0.554·8-s + 0.333·9-s − 1.34·10-s − 0.916·11-s + 0.320·12-s + 1.02·13-s + 0.242·14-s + 0.621·15-s − 1.24·16-s − 0.730·17-s − 0.415·18-s + 1.00·19-s + 0.597·20-s − 0.112·21-s + 1.14·22-s − 0.298·23-s + 0.320·24-s + 0.158·25-s − 1.27·26-s + 0.192·27-s − 0.107·28-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 \( 1 + 112.T + 8.19e3T^{2} \)
5 \( 1 - 3.76e4T + 1.22e9T^{2} \)
7 \( 1 + 6.04e4T + 9.68e10T^{2} \)
11 \( 1 + 5.38e6T + 3.45e13T^{2} \)
13 \( 1 - 1.78e7T + 3.02e14T^{2} \)
17 \( 1 + 7.26e7T + 9.90e15T^{2} \)
19 \( 1 - 2.05e8T + 4.20e16T^{2} \)
23 \( 1 + 2.11e8T + 5.04e17T^{2} \)
29 \( 1 - 4.90e9T + 1.02e19T^{2} \)
31 \( 1 + 5.76e9T + 2.44e19T^{2} \)
37 \( 1 - 5.09e8T + 2.43e20T^{2} \)
41 \( 1 + 4.36e9T + 9.25e20T^{2} \)
43 \( 1 + 1.35e9T + 1.71e21T^{2} \)
47 \( 1 + 1.26e11T + 5.46e21T^{2} \)
53 \( 1 + 2.89e10T + 2.60e22T^{2} \)
61 \( 1 - 2.30e11T + 1.61e23T^{2} \)
67 \( 1 + 1.70e11T + 5.48e23T^{2} \)
71 \( 1 + 1.48e12T + 1.16e24T^{2} \)
73 \( 1 + 1.87e12T + 1.67e24T^{2} \)
79 \( 1 + 2.52e12T + 4.66e24T^{2} \)
83 \( 1 + 1.34e12T + 8.87e24T^{2} \)
89 \( 1 + 1.41e12T + 2.19e25T^{2} \)
97 \( 1 - 1.48e13T + 6.73e25T^{2} \)
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

−9.788511651241524618112570133389, −8.884940668368815404950042348044, −8.182210997847131772634082475486, −7.11458093920980395998755055237, −5.97645668553867525723310364092, −4.70838414644884237389245645202, −3.15545658331977947525732615051, −2.02452971237693917649156158407, −1.25022662048532349888933592876, 0, 1.25022662048532349888933592876, 2.02452971237693917649156158407, 3.15545658331977947525732615051, 4.70838414644884237389245645202, 5.97645668553867525723310364092, 7.11458093920980395998755055237, 8.182210997847131772634082475486, 8.884940668368815404950042348044, 9.788511651241524618112570133389

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