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Properties

Label 2-1638-13.9-c1-0-2
Degree $2$
Conductor $1638$
Sign $0.128 - 0.991i$
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 − 1.56·5-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.780 + 1.35i)10-s + (−1.28 + 2.21i)11-s + (−0.5 − 3.57i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (0.0615 + 0.106i)17-s + (1.28 + 2.21i)19-s + (0.780 + 1.35i)20-s + (1.28 + 2.21i)22-s + (−0.561 + 0.972i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.698·5-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (−0.246 + 0.427i)10-s + (−0.386 + 0.668i)11-s + (−0.138 − 0.990i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (0.0149 + 0.0258i)17-s + (0.293 + 0.508i)19-s + (0.174 + 0.302i)20-s + (0.273 + 0.472i)22-s + (−0.117 + 0.202i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.128 - 0.991i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.128 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5142472414\)
\(L(\frac12)\) \(\approx\) \(0.5142472414\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 3.57i)T \)
good5 \( 1 + 1.56T + 5T^{2} \)
11 \( 1 + (1.28 - 2.21i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.0615 - 0.106i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.28 - 2.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.561 - 0.972i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.06 - 5.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-1.21 + 2.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.62 - 9.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.315T + 47T^{2} \)
53 \( 1 + 7T + 53T^{2} \)
59 \( 1 + (-2.56 - 4.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.62 - 9.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.56 - 7.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.68 + 9.84i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.43T + 73T^{2} \)
79 \( 1 + 6.56T + 79T^{2} \)
83 \( 1 - 5.12T + 83T^{2} \)
89 \( 1 + (-1.71 + 2.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.438 - 0.759i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.872462708544458741920226752274, −8.840414452833767290331075576361, −7.81636622028514292736334720491, −7.40483465518611225301614731258, −6.19742267115986049732217095851, −5.30127145324480245566499110128, −4.44791380782815948558392714458, −3.57676029877861558621801025528, −2.76579457277474060468395235495, −1.37520253623253776125545063749, 0.17747955018608969153414201622, 2.21936010673044061907192019995, 3.42195206903390873983348987343, 4.17202646854087974721956256394, 5.13735850246091833563259639372, 5.95949465329718796961337749226, 6.81025280196852982567004724472, 7.55786541306273233125747034165, 8.305508194747435141361644310196, 9.019519839470879636468395437352

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