Properties

Label 2-177-1.1-c13-0-83
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  + 23.4·2-s + 729·3-s − 7.64e3·4-s − 4.29e4·5-s + 1.71e4·6-s + 4.65e5·7-s − 3.71e5·8-s + 5.31e5·9-s − 1.00e6·10-s − 5.09e6·11-s − 5.57e6·12-s − 1.16e7·13-s + 1.09e7·14-s − 3.13e7·15-s + 5.38e7·16-s − 1.31e8·17-s + 1.24e7·18-s + 3.44e8·19-s + 3.28e8·20-s + 3.39e8·21-s − 1.19e8·22-s + 8.75e8·23-s − 2.70e8·24-s + 6.23e8·25-s − 2.72e8·26-s + 3.87e8·27-s − 3.55e9·28-s + ⋯
L(s)  = 1  + 0.259·2-s + 0.577·3-s − 0.932·4-s − 1.22·5-s + 0.149·6-s + 1.49·7-s − 0.501·8-s + 0.333·9-s − 0.318·10-s − 0.866·11-s − 0.538·12-s − 0.667·13-s + 0.387·14-s − 0.709·15-s + 0.802·16-s − 1.32·17-s + 0.0864·18-s + 1.67·19-s + 1.14·20-s + 0.863·21-s − 0.224·22-s + 1.23·23-s − 0.289·24-s + 0.510·25-s − 0.173·26-s + 0.192·27-s − 1.39·28-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 \( 1 - 23.4T + 8.19e3T^{2} \)
5 \( 1 + 4.29e4T + 1.22e9T^{2} \)
7 \( 1 - 4.65e5T + 9.68e10T^{2} \)
11 \( 1 + 5.09e6T + 3.45e13T^{2} \)
13 \( 1 + 1.16e7T + 3.02e14T^{2} \)
17 \( 1 + 1.31e8T + 9.90e15T^{2} \)
19 \( 1 - 3.44e8T + 4.20e16T^{2} \)
23 \( 1 - 8.75e8T + 5.04e17T^{2} \)
29 \( 1 - 1.80e9T + 1.02e19T^{2} \)
31 \( 1 + 4.43e9T + 2.44e19T^{2} \)
37 \( 1 - 2.28e10T + 2.43e20T^{2} \)
41 \( 1 + 2.03e10T + 9.25e20T^{2} \)
43 \( 1 + 2.91e10T + 1.71e21T^{2} \)
47 \( 1 + 1.00e10T + 5.46e21T^{2} \)
53 \( 1 + 9.81e10T + 2.60e22T^{2} \)
61 \( 1 - 6.23e11T + 1.61e23T^{2} \)
67 \( 1 - 9.29e10T + 5.48e23T^{2} \)
71 \( 1 - 8.86e11T + 1.16e24T^{2} \)
73 \( 1 - 5.94e11T + 1.67e24T^{2} \)
79 \( 1 + 1.79e12T + 4.66e24T^{2} \)
83 \( 1 - 6.61e10T + 8.87e24T^{2} \)
89 \( 1 + 2.43e12T + 2.19e25T^{2} \)
97 \( 1 - 8.12e12T + 6.73e25T^{2} \)
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

−9.666611739697829701599102838126, −8.604204853151723595820297657501, −7.965287258461326992887720847642, −7.24642205704936311890678733610, −5.12991963277705198100096940101, −4.73327721877430648141555179597, −3.69357605320598498819178213907, −2.58376338772635386903941279732, −1.09596995878708173717524687149, 0, 1.09596995878708173717524687149, 2.58376338772635386903941279732, 3.69357605320598498819178213907, 4.73327721877430648141555179597, 5.12991963277705198100096940101, 7.24642205704936311890678733610, 7.965287258461326992887720847642, 8.604204853151723595820297657501, 9.666611739697829701599102838126

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