Properties

Label 2-1638-39.8-c1-0-8
Degree $2$
Conductor $1638$
Sign $0.159 - 0.987i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.162 − 0.162i)5-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + 0.230i·10-s + (4.08 + 4.08i)11-s + (−2.63 + 2.45i)13-s + 1.00i·14-s − 1.00·16-s + 5.67·17-s + (−0.614 − 0.614i)19-s + (−0.162 − 0.162i)20-s − 5.77·22-s − 3.92·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.0728 − 0.0728i)5-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + 0.0728i·10-s + (1.23 + 1.23i)11-s + (−0.732 + 0.681i)13-s + 0.267i·14-s − 0.250·16-s + 1.37·17-s + (−0.141 − 0.141i)19-s + (−0.0364 − 0.0364i)20-s − 1.23·22-s − 0.817·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.159 - 0.987i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.350018302\)
\(L(\frac12)\) \(\approx\) \(1.350018302\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (2.63 - 2.45i)T \)
good5 \( 1 + (-0.162 + 0.162i)T - 5iT^{2} \)
11 \( 1 + (-4.08 - 4.08i)T + 11iT^{2} \)
17 \( 1 - 5.67T + 17T^{2} \)
19 \( 1 + (0.614 + 0.614i)T + 19iT^{2} \)
23 \( 1 + 3.92T + 23T^{2} \)
29 \( 1 + 6.30iT - 29T^{2} \)
31 \( 1 + (3.79 + 3.79i)T + 31iT^{2} \)
37 \( 1 + (1.73 - 1.73i)T - 37iT^{2} \)
41 \( 1 + (-1.60 + 1.60i)T - 41iT^{2} \)
43 \( 1 - 1.74iT - 43T^{2} \)
47 \( 1 + (-8.71 - 8.71i)T + 47iT^{2} \)
53 \( 1 - 9.16iT - 53T^{2} \)
59 \( 1 + (-4.02 - 4.02i)T + 59iT^{2} \)
61 \( 1 + 1.44T + 61T^{2} \)
67 \( 1 + (-1.99 - 1.99i)T + 67iT^{2} \)
71 \( 1 + (5.82 - 5.82i)T - 71iT^{2} \)
73 \( 1 + (-0.662 + 0.662i)T - 73iT^{2} \)
79 \( 1 - 2.96T + 79T^{2} \)
83 \( 1 + (-7.73 + 7.73i)T - 83iT^{2} \)
89 \( 1 + (-9.62 - 9.62i)T + 89iT^{2} \)
97 \( 1 + (-5.78 - 5.78i)T + 97iT^{2} \)
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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

−9.548042171658868795759881658771, −8.923704219309562640506294311296, −7.59382834456481178033050563419, −7.48996169707993311948661426586, −6.45249966398550392869302808112, −5.63214128256185524586181321727, −4.57180379373419319843383852099, −3.89153804525805378671260747768, −2.23313828650939674533403208230, −1.23760612851350604389682403046, 0.70129334947078685273581560124, 1.92095846429714064631067015953, 3.17618002623370631377204557555, 3.80893878442091919777045932312, 5.15379032470711051907553262091, 5.93121346029357798122895278875, 6.90464549757022875216097570976, 7.84009948222310686925103474382, 8.526033295378960891385827714438, 9.155426651508981307302934847180

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