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Properties

Label 2-1638-91.30-c1-0-31
Degree $2$
Conductor $1638$
Sign $0.595 + 0.803i$
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 + (0.594 + 0.343i)5-s + (2.63 − 0.246i)7-s + 0.999i·8-s − 0.686·10-s − 4.38i·11-s + (3.21 − 1.62i)13-s + (−2.15 + 1.53i)14-s + (−0.5 − 0.866i)16-s + (−2.03 + 3.52i)17-s − 7.17i·19-s + (0.594 − 0.343i)20-s + (2.19 + 3.79i)22-s + (−0.862 − 1.49i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.265 + 0.153i)5-s + (0.995 − 0.0931i)7-s + 0.353i·8-s − 0.216·10-s − 1.32i·11-s + (0.892 − 0.451i)13-s + (−0.576 + 0.409i)14-s + (−0.125 − 0.216i)16-s + (−0.493 + 0.854i)17-s − 1.64i·19-s + (0.132 − 0.0767i)20-s + (0.467 + 0.809i)22-s + (−0.179 − 0.311i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.595 + 0.803i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.358333994\)
\(L(\frac12)\) \(\approx\) \(1.358333994\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 + 0.246i)T \)
13 \( 1 + (-3.21 + 1.62i)T \)
good5 \( 1 + (-0.594 - 0.343i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 4.38iT - 11T^{2} \)
17 \( 1 + (2.03 - 3.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.17iT - 19T^{2} \)
23 \( 1 + (0.862 + 1.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.181 + 0.313i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.49 + 2.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.49 - 3.17i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.74 + 3.31i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.38 + 5.41i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.12 - 1.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.16 - 1.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.04T + 61T^{2} \)
67 \( 1 - 3.84iT - 67T^{2} \)
71 \( 1 + (-13.3 + 7.70i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.0 - 5.77i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.43 + 2.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.79iT - 83T^{2} \)
89 \( 1 + (-7.10 + 4.10i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.62 + 1.51i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.945697612750362330062875284441, −8.448361217067654140268772242029, −7.982760345231440900492397154033, −6.78322158150207424659024037765, −6.17321629367493870287364311547, −5.33633133680668893235535778714, −4.34897883821798638595073552134, −3.11212226458109212737792544342, −1.90675905387541698379165629868, −0.66060456976266482474524688186, 1.49999308013338606952961222747, 2.00291358582556837409446822881, 3.49589060540764840572671890396, 4.47883881335602843314624638840, 5.32098511427564609787691118310, 6.41065058787674029153271016120, 7.32819262855751746562712858605, 7.995836530480044950906015880120, 8.770652214873403906898902083761, 9.518373942679972872509093863854

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