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Properties

Label 2-2028-13.3-c1-0-8
Degree 22
Conductor 20282028
Sign 0.872−0.488i0.872 - 0.488i
Analytic cond. 16.193616.1936
Root an. cond. 4.024134.02413
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(2028s/2 ΓC(s) L(s)=((0.872−0.488i) Λ‾(2−s)\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}
Λ(s)=(2028s/2 ΓC(s+1/2) L(s)=((0.872−0.488i) Λ‾(1−s)\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: 22
Conductor: 20282028    =    22⋅3⋅1322^{2} \cdot 3 \cdot 13^{2}
Sign: 0.872−0.488i0.872 - 0.488i
Analytic conductor: 16.193616.1936
Root analytic conductor: 4.024134.02413
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2028(2005,⋅)\chi_{2028} (2005, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2028, ( :1/2), 0.872−0.488i)(2,\ 2028,\ (\ :1/2),\ 0.872 - 0.488i)

Particular Values

L(1)L(1) ≈\approx 1.3221617621.322161762
L(12)L(\frac12) ≈\approx 1.3221617621.322161762
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1 1
good5 1+5T2 1 + 5T^{2}
7 1+(1−1.73i)T+(−3.5−6.06i)T2 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2}
11 1+(−5.5+9.52i)T2 1 + (-5.5 + 9.52i)T^{2}
17 1+(−3+5.19i)T+(−8.5−14.7i)T2 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2}
19 1+(1−1.73i)T+(−9.5−16.4i)T2 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(−3−5.19i)T+(−14.5+25.1i)T2 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1−2T+31T2 1 - 2T + 31T^{2}
37 1+(1+1.73i)T+(−18.5+32.0i)T2 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1+(−6−10.3i)T+(−20.5+35.5i)T2 1 + (-6 - 10.3i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2+3.46i)T+(−21.5−37.2i)T2 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2}
47 1+47T2 1 + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+(6−10.3i)T+(−29.5−51.0i)T2 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2}
61 1+(1−1.73i)T+(−30.5−52.8i)T2 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2}
67 1+(−5−8.66i)T+(−33.5+58.0i)T2 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2}
71 1+(6−10.3i)T+(−35.5−61.4i)T2 1 + (6 - 10.3i)T + (-35.5 - 61.4i)T^{2}
73 1−14T+73T2 1 - 14T + 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1+(−44.5+77.0i)T2 1 + (-44.5 + 77.0i)T^{2}
97 1+(−5+8.66i)T+(−48.5−84.0i)T2 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line