↑

Properties

Label 2-1386-77.10-c1-0-17
Degree $2$
Conductor $1386$
Sign $0.679 + 0.734i$
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 + (−3.71 + 2.14i)5-s + (0.293 − 2.62i)7-s − 0.999i·8-s + (−2.14 + 3.71i)10-s + (−1.03 + 3.15i)11-s + 4.87·13-s + (−1.06 − 2.42i)14-s + (−0.5 − 0.866i)16-s + (1.58 − 2.75i)17-s + (1.05 + 1.82i)19-s + 4.28i·20-s + (0.682 + 3.24i)22-s + (−1.06 − 1.83i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.66 + 0.958i)5-s + (0.110 − 0.993i)7-s − 0.353i·8-s + (−0.677 + 1.17i)10-s + (−0.311 + 0.950i)11-s + 1.35·13-s + (−0.283 − 0.647i)14-s + (−0.125 − 0.216i)16-s + (0.385 − 0.667i)17-s + (0.241 + 0.418i)19-s + 0.958i·20-s + (0.145 + 0.691i)22-s + (−0.221 − 0.382i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.679 + 0.734i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.679 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.785853153\)
\(L(\frac12)\) \(\approx\) \(1.785853153\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.293 + 2.62i)T \)
11 \( 1 + (1.03 - 3.15i)T \)
good5 \( 1 + (3.71 - 2.14i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 4.87T + 13T^{2} \)
17 \( 1 + (-1.58 + 2.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.05 - 1.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.06 + 1.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.88iT - 29T^{2} \)
31 \( 1 + (-5.20 - 3.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.75 - 3.04i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.28T + 41T^{2} \)
43 \( 1 + 9.33iT - 43T^{2} \)
47 \( 1 + (-7.40 + 4.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.09 + 5.36i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.485 - 0.280i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.06 - 1.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.94 + 6.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.03T + 71T^{2} \)
73 \( 1 + (-0.587 + 1.01i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.1 - 5.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + (1.01 - 0.587i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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

−9.823520557602815734084355958574, −8.375960843142862337924478752210, −7.71010727894610508917570551392, −7.04766901449664549126484157538, −6.36245113766975295708392875201, −4.96532659168228958272827416404, −3.99277813576107954339498971018, −3.68846990627949094350400981626, −2.54487780395065171749132801937, −0.78206885417047754246538513331, 1.06299317958107921612441644175, 2.96502885023418990619435925781, 3.76866940753902360870662818091, 4.55294129469126267203986529159, 5.54478777386788378760215923119, 6.12311807124793128595062381583, 7.42186361145275246815114074249, 8.220540609261764744884873135282, 8.506007299300036794768745161676, 9.296097849751335886644536223939

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