Properties

Label 2-177-1.1-c13-0-62
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  − 66.2·2-s − 729·3-s − 3.79e3·4-s + 2.41e4·5-s + 4.83e4·6-s − 3.95e5·7-s + 7.94e5·8-s + 5.31e5·9-s − 1.60e6·10-s + 9.00e6·11-s + 2.76e6·12-s − 3.09e7·13-s + 2.61e7·14-s − 1.76e7·15-s − 2.15e7·16-s + 5.93e7·17-s − 3.52e7·18-s + 1.66e8·19-s − 9.18e7·20-s + 2.87e8·21-s − 5.97e8·22-s − 1.38e9·23-s − 5.79e8·24-s − 6.35e8·25-s + 2.05e9·26-s − 3.87e8·27-s + 1.49e9·28-s + ⋯
L(s)  = 1  − 0.732·2-s − 0.577·3-s − 0.463·4-s + 0.692·5-s + 0.422·6-s − 1.26·7-s + 1.07·8-s + 0.333·9-s − 0.507·10-s + 1.53·11-s + 0.267·12-s − 1.77·13-s + 0.929·14-s − 0.399·15-s − 0.321·16-s + 0.596·17-s − 0.244·18-s + 0.810·19-s − 0.320·20-s + 0.732·21-s − 1.12·22-s − 1.95·23-s − 0.618·24-s − 0.520·25-s + 1.30·26-s − 0.192·27-s + 0.588·28-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 \( 1 + 66.2T + 8.19e3T^{2} \)
5 \( 1 - 2.41e4T + 1.22e9T^{2} \)
7 \( 1 + 3.95e5T + 9.68e10T^{2} \)
11 \( 1 - 9.00e6T + 3.45e13T^{2} \)
13 \( 1 + 3.09e7T + 3.02e14T^{2} \)
17 \( 1 - 5.93e7T + 9.90e15T^{2} \)
19 \( 1 - 1.66e8T + 4.20e16T^{2} \)
23 \( 1 + 1.38e9T + 5.04e17T^{2} \)
29 \( 1 - 2.71e9T + 1.02e19T^{2} \)
31 \( 1 - 4.57e9T + 2.44e19T^{2} \)
37 \( 1 + 9.75e9T + 2.43e20T^{2} \)
41 \( 1 + 1.59e10T + 9.25e20T^{2} \)
43 \( 1 - 6.97e7T + 1.71e21T^{2} \)
47 \( 1 - 1.45e11T + 5.46e21T^{2} \)
53 \( 1 + 3.12e11T + 2.60e22T^{2} \)
61 \( 1 + 3.14e11T + 1.61e23T^{2} \)
67 \( 1 - 1.17e12T + 5.48e23T^{2} \)
71 \( 1 + 9.67e11T + 1.16e24T^{2} \)
73 \( 1 - 2.15e12T + 1.67e24T^{2} \)
79 \( 1 - 2.07e12T + 4.66e24T^{2} \)
83 \( 1 - 3.77e12T + 8.87e24T^{2} \)
89 \( 1 - 1.92e12T + 2.19e25T^{2} \)
97 \( 1 + 7.77e12T + 6.73e25T^{2} \)
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

−9.713650728571082875187604298040, −9.369212417781825663684952664244, −7.85896576104468639332615905031, −6.78117001063187732214309550752, −5.91805172928323688968596167871, −4.72798235329188960184826733356, −3.59558195689288275313830516286, −2.05929018651757460260930513305, −0.898046180279481024967707226548, 0, 0.898046180279481024967707226548, 2.05929018651757460260930513305, 3.59558195689288275313830516286, 4.72798235329188960184826733356, 5.91805172928323688968596167871, 6.78117001063187732214309550752, 7.85896576104468639332615905031, 9.369212417781825663684952664244, 9.713650728571082875187604298040

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