Properties

Label 2-177-1.1-c13-0-64
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 28.6·2-s + 729·3-s − 7.37e3·4-s + 6.00e4·5-s + 2.08e4·6-s + 4.28e5·7-s − 4.45e5·8-s + 5.31e5·9-s + 1.71e6·10-s − 5.05e6·11-s − 5.37e6·12-s + 8.30e6·13-s + 1.22e7·14-s + 4.37e7·15-s + 4.76e7·16-s − 3.21e7·17-s + 1.52e7·18-s + 9.20e7·19-s − 4.42e8·20-s + 3.12e8·21-s − 1.44e8·22-s + 2.82e8·23-s − 3.24e8·24-s + 2.38e9·25-s + 2.37e8·26-s + 3.87e8·27-s − 3.15e9·28-s + ⋯
L(s)  = 1  + 0.316·2-s + 0.577·3-s − 0.900·4-s + 1.71·5-s + 0.182·6-s + 1.37·7-s − 0.600·8-s + 0.333·9-s + 0.543·10-s − 0.859·11-s − 0.519·12-s + 0.477·13-s + 0.434·14-s + 0.992·15-s + 0.710·16-s − 0.322·17-s + 0.105·18-s + 0.448·19-s − 1.54·20-s + 0.793·21-s − 0.271·22-s + 0.397·23-s − 0.346·24-s + 1.95·25-s + 0.150·26-s + 0.192·27-s − 1.23·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(4.962009555\)
\(L(\frac12)\) \(\approx\) \(4.962009555\)
\(L(\frac{15}{2})\) 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 \( 1 - 28.6T + 8.19e3T^{2} \)
5 \( 1 - 6.00e4T + 1.22e9T^{2} \)
7 \( 1 - 4.28e5T + 9.68e10T^{2} \)
11 \( 1 + 5.05e6T + 3.45e13T^{2} \)
13 \( 1 - 8.30e6T + 3.02e14T^{2} \)
17 \( 1 + 3.21e7T + 9.90e15T^{2} \)
19 \( 1 - 9.20e7T + 4.20e16T^{2} \)
23 \( 1 - 2.82e8T + 5.04e17T^{2} \)
29 \( 1 + 1.86e9T + 1.02e19T^{2} \)
31 \( 1 + 1.29e9T + 2.44e19T^{2} \)
37 \( 1 + 5.91e9T + 2.43e20T^{2} \)
41 \( 1 - 4.67e10T + 9.25e20T^{2} \)
43 \( 1 - 5.00e10T + 1.71e21T^{2} \)
47 \( 1 - 5.50e10T + 5.46e21T^{2} \)
53 \( 1 + 5.12e10T + 2.60e22T^{2} \)
61 \( 1 - 4.21e11T + 1.61e23T^{2} \)
67 \( 1 + 6.77e9T + 5.48e23T^{2} \)
71 \( 1 - 6.84e11T + 1.16e24T^{2} \)
73 \( 1 + 1.74e12T + 1.67e24T^{2} \)
79 \( 1 + 4.15e12T + 4.66e24T^{2} \)
83 \( 1 - 1.01e12T + 8.87e24T^{2} \)
89 \( 1 + 4.21e12T + 2.19e25T^{2} \)
97 \( 1 + 3.08e12T + 6.73e25T^{2} \)
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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

−10.18374043432452673001887962938, −9.236954254963797357097021459286, −8.588733191138128259478108768530, −7.49367487351830048581576609423, −5.85281149375063562751705653060, −5.24577106347655488742999216615, −4.29281058649650045554737948497, −2.81705224412214429376779576017, −1.87828664533289205259380790556, −0.932097047379386479517097638137, 0.932097047379386479517097638137, 1.87828664533289205259380790556, 2.81705224412214429376779576017, 4.29281058649650045554737948497, 5.24577106347655488742999216615, 5.85281149375063562751705653060, 7.49367487351830048581576609423, 8.588733191138128259478108768530, 9.236954254963797357097021459286, 10.18374043432452673001887962938

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