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Properties

Label 2-1638-91.30-c1-0-29
Degree $2$
Conductor $1638$
Sign $-0.570 + 0.821i$
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 + (0.152 + 0.0882i)5-s + (−0.623 − 2.57i)7-s + 0.999i·8-s − 0.176·10-s + 2.92i·11-s + (−2.87 + 2.18i)13-s + (1.82 + 1.91i)14-s + (−0.5 − 0.866i)16-s + (0.536 − 0.928i)17-s − 4.80i·19-s + (0.152 − 0.0882i)20-s + (−1.46 − 2.53i)22-s + (0.966 + 1.67i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.0683 + 0.0394i)5-s + (−0.235 − 0.971i)7-s + 0.353i·8-s − 0.0558·10-s + 0.882i·11-s + (−0.796 + 0.604i)13-s + (0.487 + 0.511i)14-s + (−0.125 − 0.216i)16-s + (0.130 − 0.225i)17-s − 1.10i·19-s + (0.0341 − 0.0197i)20-s + (−0.311 − 0.540i)22-s + (0.201 + 0.348i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.570 + 0.821i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.570 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4491833893\)
\(L(\frac12)\) \(\approx\) \(0.4491833893\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.623 + 2.57i)T \)
13 \( 1 + (2.87 - 2.18i)T \)
good5 \( 1 + (-0.152 - 0.0882i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.92iT - 11T^{2} \)
17 \( 1 + (-0.536 + 0.928i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 4.80iT - 19T^{2} \)
23 \( 1 + (-0.966 - 1.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.880 - 1.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.771 + 0.445i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.26 + 3.61i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.65 - 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.42 + 4.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.56 + 3.79i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.98 + 6.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.3 + 7.15i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.40T + 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 + (11.6 - 6.71i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.20 - 5.31i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.86 + 6.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.49iT - 83T^{2} \)
89 \( 1 + (4.92 - 2.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.29 + 5.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.335516470015316992098976984610, −8.229488955598757125725772729636, −7.30759083330515183854072085255, −7.03456631360041777738259290374, −6.10055628942695965937108948404, −4.87394286028535560418454919841, −4.27062173416047833415626346772, −2.86401882181294244859726878079, −1.70776201177492350952454511034, −0.20912241910512052984230787332, 1.45609986910436818156404878685, 2.70216276429857303540083536410, 3.36499101739129645135051275252, 4.70109954409170495846024006860, 5.81967810144491317853439306426, 6.26183721177521187512963107732, 7.67293428852804961938456097708, 8.034930175283109593569156161578, 9.032014938570932340438688995343, 9.535851448308526556584463578271

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