↑

Properties

Label 2-3087-49.17-c0-0-0
Degree $2$
Conductor $3087$
Sign $0.972 - 0.234i$
Analytic cond. $1.54061$
Root an. cond. $1.24121$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.988 − 0.149i)4-s + (−0.846 + 1.75i)13-s + (0.955 − 0.294i)16-s + (0.751 − 0.433i)19-s + (0.0747 − 0.997i)25-s + (1.35 + 0.781i)31-s + (1.23 + 0.185i)37-s + (−0.400 + 1.75i)43-s + (−0.574 + 1.86i)52-s + (0.233 − 1.54i)61-s + (0.900 − 0.433i)64-s + (0.222 − 0.385i)67-s + (−1.55 − 0.116i)73-s + (0.678 − 0.541i)76-s + (−0.623 − 1.07i)79-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)4-s + (−0.846 + 1.75i)13-s + (0.955 − 0.294i)16-s + (0.751 − 0.433i)19-s + (0.0747 − 0.997i)25-s + (1.35 + 0.781i)31-s + (1.23 + 0.185i)37-s + (−0.400 + 1.75i)43-s + (−0.574 + 1.86i)52-s + (0.233 − 1.54i)61-s + (0.900 − 0.433i)64-s + (0.222 − 0.385i)67-s + (−1.55 − 0.116i)73-s + (0.678 − 0.541i)76-s + (−0.623 − 1.07i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3087\)    =    \(3^{2} \cdot 7^{3}\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(1.54061\)
Root analytic conductor: \(1.24121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3087} (2971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3087,\ (\ :0),\ 0.972 - 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.602489328\)
\(L(\frac12)\) \(\approx\) \(1.602489328\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.988 + 0.149i)T^{2} \)
5 \( 1 + (-0.0747 + 0.997i)T^{2} \)
11 \( 1 + (0.365 - 0.930i)T^{2} \)
13 \( 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (-0.751 + 0.433i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (-1.35 - 0.781i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.23 - 0.185i)T + (0.955 + 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.400 - 1.75i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.988 - 0.149i)T^{2} \)
53 \( 1 + (0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (-0.233 + 1.54i)T + (-0.955 - 0.294i)T^{2} \)
67 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (1.55 + 0.116i)T + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
97 \( 1 + 0.867iT - T^{2} \)
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

−8.968462437880279850764209651796, −8.015434267703842421777525728870, −7.33789495265213442747011702707, −6.57385836888950413292422425550, −6.20464249851163741770121568050, −4.95459007486392267136265686484, −4.38785751157423463735587716700, −3.08326387540440846630394589384, −2.37442514326450320999576569180, −1.35476252420495520108719072637, 1.11891633270732890297041529812, 2.48348547066613168651994531683, 3.03050008038358042964753640187, 4.02163489338580366555709155135, 5.32207130034531386495442017319, 5.69901053392058327345309170455, 6.67850895289388844476356733464, 7.55504753371658424123315069385, 7.79879575907585497243019899117, 8.728975832214837345869291178511

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