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Properties

Label 2-2028-13.9-c1-0-12
Degree $2$
Conductor $2028$
Sign $0.872 + 0.488i$
Analytic cond. $16.1936$
Root an. cond. $4.02413$
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)3-s + (−1 − 1.73i)7-s + (−0.499 − 0.866i)9-s + (3 + 5.19i)17-s + (−1 − 1.73i)19-s + 1.99·21-s − 5·25-s + 0.999·27-s + (3 − 5.19i)29-s + 2·31-s + (−1 + 1.73i)37-s + (6 − 10.3i)41-s + (2 + 3.46i)43-s + (1.50 − 2.59i)49-s − 6·51-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.377 − 0.654i)7-s + (−0.166 − 0.288i)9-s + (0.727 + 1.26i)17-s + (−0.229 − 0.397i)19-s + 0.436·21-s − 25-s + 0.192·27-s + (0.557 − 0.964i)29-s + 0.359·31-s + (−0.164 + 0.284i)37-s + (0.937 − 1.62i)41-s + (0.304 + 0.528i)43-s + (0.214 − 0.371i)49-s − 0.840·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(16.1936\)
Root analytic conductor: \(4.02413\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.322161762\)
\(L(\frac12)\) \(\approx\) \(1.322161762\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good5 \( 1 + 5T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6 + 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.260358535321253297364560631877, −8.229923262473455737588904134127, −7.62380026856888269742888722969, −6.53862175127387055651464783190, −5.99346589870274352267000065110, −5.01123426105569227446842105466, −4.04187562692168797128792034790, −3.49523214166273476029267499505, −2.12979591817287860065275745328, −0.60913896169107773097867352558, 1.00085319178041589144924190940, 2.34823047376493326616499330801, 3.16782299344602837204615425987, 4.39319859747440900322996887418, 5.42365268598637903801084274300, 5.96528201641067534523912530669, 6.87680229037414437533032846336, 7.59011098638213894290419386613, 8.374440196046065466052426888602, 9.243760010912568902037591365533

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