↑

Properties

Label 2-1638-91.81-c1-0-3
Degree $2$
Conductor $1638$
Sign $-0.740 - 0.671i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.97 + 3.42i)5-s + (−1.48 + 2.18i)7-s − 0.999·8-s + 3.95·10-s − 4.91·11-s + (−3.39 − 1.21i)13-s + (1.15 + 2.38i)14-s + (−0.5 + 0.866i)16-s + (0.0702 + 0.121i)17-s − 0.776·19-s + (1.97 − 3.42i)20-s + (−2.45 + 4.25i)22-s + (4.76 − 8.25i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.883 + 1.52i)5-s + (−0.561 + 0.827i)7-s − 0.353·8-s + 1.24·10-s − 1.48·11-s + (−0.941 − 0.338i)13-s + (0.307 + 0.636i)14-s + (−0.125 + 0.216i)16-s + (0.0170 + 0.0295i)17-s − 0.178·19-s + (0.441 − 0.764i)20-s + (−0.523 + 0.907i)22-s + (0.994 − 1.72i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.740 - 0.671i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.740 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7293362561\)
\(L(\frac12)\) \(\approx\) \(0.7293362561\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (1.48 - 2.18i)T \)
13 \( 1 + (3.39 + 1.21i)T \)
good5 \( 1 + (-1.97 - 3.42i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 4.91T + 11T^{2} \)
17 \( 1 + (-0.0702 - 0.121i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 0.776T + 19T^{2} \)
23 \( 1 + (-4.76 + 8.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.629 - 1.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.67 - 2.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.56 - 9.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.65 + 8.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.541 - 0.937i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.33 + 5.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.53 - 9.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.215 - 0.373i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.28T + 61T^{2} \)
67 \( 1 - 8.19T + 67T^{2} \)
71 \( 1 + (1.93 - 3.35i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.0817 - 0.141i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.17 - 3.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (0.536 - 0.929i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.54 - 11.3i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.00468583348090634565026671170, −9.145015485948585928489055524838, −8.166573979579142558141388566910, −6.97478969633976740062147253926, −6.49310688014655320209191697425, −5.47258644460366059163891399930, −4.95088214422533333512832078412, −3.22848983568010668701760267079, −2.75399287474828974674227772555, −2.13705916819401050220073148928, 0.22141033114732812454576925114, 1.77767944174787153497943066335, 3.11750101373568749863973045128, 4.35708143471029290059597033139, 5.13861049855919406007344248081, 5.52786059651704292717072363767, 6.62826591847741583363069450103, 7.55066915778454125503411234391, 8.101424531016637350823964028473, 9.256722870304320112813338636698

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