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Properties

Label 2-1638-91.25-c1-0-10
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

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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: $\chi_{1638} (1117, \cdot )$
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 \( 1 + 16.1iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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