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Properties

Label 2-2888-152.123-c0-0-3
Degree 22
Conductor 28882888
Sign 0.714+0.700i0.714 + 0.700i
Analytic cond. 1.441291.44129
Root an. cond. 1.200541.20054
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.766 − 0.642i)16-s + (0.347 + 1.96i)17-s + (0.766 − 0.642i)22-s + (0.939 + 0.342i)24-s + (0.766 + 0.642i)25-s + (0.499 + 0.866i)27-s + (−0.766 − 0.642i)32-s + (0.939 + 0.342i)33-s + (1.87 − 0.684i)34-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + (0.766 − 0.642i)16-s + (0.347 + 1.96i)17-s + (0.766 − 0.642i)22-s + (0.939 + 0.342i)24-s + (0.766 + 0.642i)25-s + (0.499 + 0.866i)27-s + (−0.766 − 0.642i)32-s + (0.939 + 0.342i)33-s + (1.87 − 0.684i)34-s + ⋯

Functional equation

Λ(s)=(2888s/2 ΓC(s) L(s)=((0.714+0.700i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2888s/2 ΓC(s) L(s)=((0.714+0.700i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28882888    =    23⋅1922^{3} \cdot 19^{2}
Sign: 0.714+0.700i0.714 + 0.700i
Analytic conductor: 1.441291.44129
Root analytic conductor: 1.200541.20054
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2888(2555,⋅)\chi_{2888} (2555, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2888, ( :0), 0.714+0.700i)(2,\ 2888,\ (\ :0),\ 0.714 + 0.700i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3515480201.351548020
L(12)L(\frac12) ≈\approx 1.3515480201.351548020
L(1)L(1) 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+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
19 1 1
good3 1+(−0.766+0.642i)T+(0.173−0.984i)T2 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2}
5 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
17 1+(−0.347−1.96i)T+(−0.939+0.342i)T2 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2}
23 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
29 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(−0.766+0.642i)T+(0.173−0.984i)T2 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2}
43 1+(1.87+0.684i)T+(0.766+0.642i)T2 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2}
47 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
53 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
59 1+(−0.173−0.984i)T+(−0.939+0.342i)T2 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2}
61 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
67 1+(−0.173+0.984i)T+(−0.939−0.342i)T2 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}
71 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
73 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
79 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
83 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
89 1+(1.53+1.28i)T+(0.173+0.984i)T2 1 + (1.53 + 1.28i)T + (0.173 + 0.984i)T^{2}
97 1+(−0.173−0.984i)T+(−0.939+0.342i)T2 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)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

−8.693969863280707446424814144264, −8.404576939099660407216881528732, −7.54119766300800202011269890721, −6.85804750724784958760856829780, −5.69589936947344784667275560320, −4.72635166502805319303727712921, −3.83063502472055764933929353401, −3.09160034479100497144954793305, −1.96394389029809764817339897498, −1.51130941652604428453138521343, 0.923645274946067624005142048956, 2.82641898652551433707878187510, 3.50341200695774901342190382091, 4.50151199209770002992900826768, 5.11028872863095017786895798927, 6.15434123755423510434533787128, 6.76274923356974620945095950224, 7.64878694315030921847091771969, 8.414299334865891283984789439087, 8.915282386766325044860552216063

Graph of the ZZ-function along the critical line