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Properties

Label 2-1638-7.2-c1-0-27
Degree $2$
Conductor $1638$
Sign $0.0633 + 0.997i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.82 + 3.15i)5-s + (1.32 − 2.29i)7-s + 0.999·8-s + (−1.82 − 3.15i)10-s + (0.322 + 0.559i)11-s + 13-s + (1.32 + 2.29i)14-s + (−0.5 + 0.866i)16-s + (−3.32 − 5.75i)17-s + (−2.5 + 4.33i)19-s + 3.64·20-s − 0.645·22-s + (−1.17 + 2.03i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.815 + 1.41i)5-s + (0.499 − 0.866i)7-s + 0.353·8-s + (−0.576 − 0.998i)10-s + (0.0973 + 0.168i)11-s + 0.277·13-s + (0.353 + 0.612i)14-s + (−0.125 + 0.216i)16-s + (−0.805 − 1.39i)17-s + (−0.573 + 0.993i)19-s + 0.815·20-s − 0.137·22-s + (−0.245 + 0.425i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2775609447\)
\(L(\frac12)\) \(\approx\) \(0.2775609447\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-1.32 + 2.29i)T \)
13 \( 1 - T \)
good5 \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.322 - 0.559i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.32 + 5.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.17 - 2.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.29T + 29T^{2} \)
31 \( 1 + (1.64 + 2.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.82 - 4.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.35T + 41T^{2} \)
43 \( 1 + 5.29T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.96 + 6.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.96 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.79 + 6.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 16.2T + 71T^{2} \)
73 \( 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (-8.46 + 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.937T + 97T^{2} \)
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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

−9.144860580889699363349714422004, −8.022303984563423528162673010792, −7.62182470498022745892343404528, −6.88147249201879466069125056572, −6.34842504328708911622679400679, −5.06884078556900675715565155716, −4.09494275655900844671445158349, −3.34510457207087140901389637118, −1.93377480380183226901149110161, −0.12496992900979926841914929406, 1.32305886670911504502077320400, 2.33541034715053379919063573575, 3.78333597354226173200310627106, 4.44849621464829863566020835816, 5.25636479859120463839027151465, 6.26286108072198574846384111796, 7.55040138354461976354855034778, 8.328842694756239380556452124294, 8.899525553049583262740067327998, 9.097603598975220011355775566379

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