↑

Properties

Label 2-1638-91.51-c1-0-46
Degree $2$
Conductor $1638$
Sign $0.240 + 0.970i$
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 + (−0.294 − 0.169i)5-s + (0.420 − 2.61i)7-s + 0.999i·8-s + (−0.169 − 0.294i)10-s + (0.571 − 0.330i)11-s + (0.660 − 3.54i)13-s + (1.66 − 2.05i)14-s + (−0.5 + 0.866i)16-s + (−3.27 − 5.66i)17-s + (−5.27 − 3.04i)19-s − 0.339i·20-s + 0.660·22-s + (−3.71 + 6.43i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.131 − 0.0759i)5-s + (0.158 − 0.987i)7-s + 0.353i·8-s + (−0.0537 − 0.0930i)10-s + (0.172 − 0.0995i)11-s + (0.183 − 0.983i)13-s + (0.446 − 0.548i)14-s + (−0.125 + 0.216i)16-s + (−0.793 − 1.37i)17-s + (−1.20 − 0.698i)19-s − 0.0759i·20-s + 0.140·22-s + (−0.774 + 1.34i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.240 + 0.970i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.240 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.809298141\)
\(L(\frac12)\) \(\approx\) \(1.809298141\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.420 + 2.61i)T \)
13 \( 1 + (-0.660 + 3.54i)T \)
good5 \( 1 + (0.294 + 0.169i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.571 + 0.330i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.27 + 5.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.27 + 3.04i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.71 - 6.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.56T + 29T^{2} \)
31 \( 1 + (-3.46 + 2i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.06 + 1.77i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.864iT - 41T^{2} \)
43 \( 1 + 5.08T + 43T^{2} \)
47 \( 1 + (-8.18 - 4.72i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.38 + 1.95i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.932 - 1.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.45 + 3.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.884iT - 71T^{2} \)
73 \( 1 + (-8.72 + 5.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.22 + 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.2iT - 83T^{2} \)
89 \( 1 + (3.85 + 2.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.1iT - 97T^{2} \)
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

−9.117176339651219652897369867896, −8.192665773674632495395269328024, −7.55600349741664224825157666652, −6.77531517350130524530813040885, −6.01663462071287009135373577621, −4.93677548610591519176205347954, −4.29757154778344214004298001584, −3.39119078836661350860954142230, −2.25898978651422428438905230927, −0.53389473310578936467946093810, 1.76127898611439442804718193400, 2.40338515878616116806813531379, 3.81733526182766304205288585873, 4.36353804688330375974993719552, 5.43686468373993373678454142948, 6.36832152566709628690340340426, 6.71737464443858643075025136908, 8.384550026061377510153122653069, 8.492967245606354398182019224779, 9.639397193751639402423134833421

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