↑

Properties

Label 2-1386-77.54-c1-0-12
Degree $2$
Conductor $1386$
Sign $0.999 + 0.0189i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−2.83 − 1.63i)5-s + (2.54 − 0.732i)7-s − 0.999i·8-s + (1.63 + 2.83i)10-s + (2.54 + 2.12i)11-s − 5.12·13-s + (−2.56 − 0.637i)14-s + (−0.5 + 0.866i)16-s + (2.66 + 4.61i)17-s + (−2.13 + 3.70i)19-s − 3.26i·20-s + (−1.14 − 3.11i)22-s + (2.68 − 4.65i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.26 − 0.730i)5-s + (0.960 − 0.276i)7-s − 0.353i·8-s + (0.516 + 0.895i)10-s + (0.767 + 0.641i)11-s − 1.42·13-s + (−0.686 − 0.170i)14-s + (−0.125 + 0.216i)16-s + (0.645 + 1.11i)17-s + (−0.490 + 0.849i)19-s − 0.730i·20-s + (−0.243 − 0.663i)22-s + (0.560 − 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.999 + 0.0189i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.999 + 0.0189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9646713279\)
\(L(\frac12)\) \(\approx\) \(0.9646713279\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2.54 + 0.732i)T \)
11 \( 1 + (-2.54 - 2.12i)T \)
good5 \( 1 + (2.83 + 1.63i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 + (-2.66 - 4.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.13 - 3.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.68 + 4.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.77iT - 29T^{2} \)
31 \( 1 + (-5.59 + 3.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.06 - 1.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.949T + 41T^{2} \)
43 \( 1 - 6.20iT - 43T^{2} \)
47 \( 1 + (10.4 + 6.01i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.26 - 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.6 + 6.74i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.67 + 4.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.09 + 5.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + (-6.61 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.93 - 4.58i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.835T + 83T^{2} \)
89 \( 1 + (-5.39 - 3.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.624iT - 97T^{2} \)
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

−9.631640512338591839462656174831, −8.446978586193188652464462142562, −8.217051390122634280670948010934, −7.43649175725560067619073212556, −6.60528400938197853612883561126, −5.04419061611836300761607338001, −4.40351113864669530082542084142, −3.62549497370899301983596950861, −2.07855897097382684779477328231, −0.951416962187597445838941187796, 0.65519193016563061727323709979, 2.36686284376217869184854191962, 3.40650254038363026310285483307, 4.61540497792440797149644488306, 5.35630765452707441305116368863, 6.69562333362730289061042744645, 7.27034732939911939187922603878, 7.88327182148257685446007224908, 8.631100606735128442916641013633, 9.470599085263339158548913771815

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