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Properties

Label 2-1638-13.4-c1-0-18
Degree $2$
Conductor $1638$
Sign $0.582 + 0.812i$
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 + 0.145i·5-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (0.0727 − 0.126i)10-s + (−1.30 − 0.754i)11-s + (−2.95 + 2.06i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (−0.160 − 0.277i)17-s + (−1.44 + 0.836i)19-s + (−0.126 + 0.0727i)20-s + (0.754 + 1.30i)22-s + (2.75 − 4.76i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.0650i·5-s + (−0.327 + 0.188i)7-s − 0.353i·8-s + (0.0230 − 0.0398i)10-s + (−0.394 − 0.227i)11-s + (−0.819 + 0.572i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (−0.0389 − 0.0674i)17-s + (−0.332 + 0.191i)19-s + (−0.0281 + 0.0162i)20-s + (0.160 + 0.278i)22-s + (0.573 − 0.994i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.582 + 0.812i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.582 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9943577431\)
\(L(\frac12)\) \(\approx\) \(0.9943577431\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.95 - 2.06i)T \)
good5 \( 1 - 0.145iT - 5T^{2} \)
11 \( 1 + (1.30 + 0.754i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.160 + 0.277i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.44 - 0.836i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.75 + 4.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.59 + 7.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.879iT - 31T^{2} \)
37 \( 1 + (-3.14 - 1.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.12 + 0.651i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.499 - 0.864i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.30iT - 47T^{2} \)
53 \( 1 - 5.12T + 53T^{2} \)
59 \( 1 + (-2.78 + 1.60i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.63 - 2.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.99 - 1.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.10 + 3.52i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.35iT - 73T^{2} \)
79 \( 1 - 8.66T + 79T^{2} \)
83 \( 1 + 0.148iT - 83T^{2} \)
89 \( 1 + (-7.13 - 4.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.4 + 6.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.287196349747015128390372336869, −8.560773465815229504966807216499, −7.83480239945357939331490949941, −6.88312975818061101312662135713, −6.27282651431392514104159426353, −5.04813250327658472885068256935, −4.17093629436783771187157811379, −2.90863578698812248330193529473, −2.23075588165742621743350954359, −0.60359883653451272847222044652, 0.941140317416486779064734856960, 2.40633072170721367297387497860, 3.39143171099447149253112455340, 4.80503775796324250928829367851, 5.37890735852945181083516885439, 6.52706890679793100273041473592, 7.16902659410606555354626620868, 7.86466442357592617926474176247, 8.766665564933825810773433158461, 9.399819440313829630280250962466

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