Properties

Label 2-1638-13.10-c1-0-3
Degree $2$
Conductor $1638$
Sign $-0.796 - 0.604i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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 + 4.39i·5-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (2.19 + 3.80i)10-s + (−0.971 + 0.560i)11-s + (−2.14 + 2.89i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (−1.57 + 2.73i)17-s + (−6.99 − 4.03i)19-s + (3.80 + 2.19i)20-s + (−0.560 + 0.971i)22-s + (2.70 + 4.68i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 1.96i·5-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (0.695 + 1.20i)10-s + (−0.292 + 0.169i)11-s + (−0.594 + 0.804i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.382 + 0.662i)17-s + (−1.60 − 0.926i)19-s + (0.851 + 0.491i)20-s + (−0.119 + 0.207i)22-s + (0.563 + 0.976i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.796 - 0.604i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.796 - 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.070248514\)
\(L(\frac12)\) \(\approx\) \(1.070248514\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (2.14 - 2.89i)T \)
good5 \( 1 - 4.39iT - 5T^{2} \)
11 \( 1 + (0.971 - 0.560i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.57 - 2.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.99 + 4.03i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.70 - 4.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.37 + 5.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.58iT - 31T^{2} \)
37 \( 1 + (-6.59 + 3.80i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.78 + 1.02i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.792 + 1.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.58iT - 47T^{2} \)
53 \( 1 + 8.44T + 53T^{2} \)
59 \( 1 + (-6.30 - 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.20 - 9.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.56 + 2.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.51 + 4.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + 3.52T + 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + (9.02 - 5.21i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.06 - 0.613i)T + (48.5 + 84.0i)T^{2} \)
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

−9.880567156300091969785628755076, −9.219206144327372631800249947154, −7.74935628452047679818017103516, −7.14829532091890092458778205483, −6.40652387971232975769541890748, −5.88771072776142760991175039458, −4.39398539137537110532858982536, −3.80738545438850390363077550142, −2.61991814842393730727331094433, −2.20237286051599991192325534386, 0.30060940435165579310297570843, 1.84333673423888684675837899759, 3.13681581693547915308390869092, 4.37934311125306222578695702599, 4.88936330048132973077232216424, 5.60836235906463141693758580226, 6.44306333541726143089780980988, 7.57031191558874764078407016712, 8.409221630383032753953881366444, 8.774512632588099849576408477361

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