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Properties

Label 2-2100-5.4-c3-0-8
Degree 22
Conductor 21002100
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 123.904123.904
Root an. cond. 11.131211.1312
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 7i·7-s − 9·9-s + 4·11-s − 54i·13-s − 14i·17-s − 92·19-s − 21·21-s + 152i·23-s + 27i·27-s + 106·29-s − 144·31-s − 12i·33-s + 158i·37-s − 162·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.109·11-s − 1.15i·13-s − 0.199i·17-s − 1.11·19-s − 0.218·21-s + 1.37i·23-s + 0.192i·27-s + 0.678·29-s − 0.834·31-s − 0.0633i·33-s + 0.702i·37-s − 0.665·39-s + ⋯

Functional equation

Λ(s)=(2100s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(2100s/2 ΓC(s+3/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21002100    =    22⋅3⋅52⋅72^{2} \cdot 3 \cdot 5^{2} \cdot 7
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 123.904123.904
Root analytic conductor: 11.131211.1312
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ2100(1849,⋅)\chi_{2100} (1849, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2100, ( :3/2), 0.447−0.894i)(2,\ 2100,\ (\ :3/2),\ 0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 0.89803272550.8980327255
L(12)L(\frac12) ≈\approx 0.89803272550.8980327255
L(52)L(\frac{5}{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+3iT 1 + 3iT
5 1 1
7 1+7iT 1 + 7iT
good11 1−4T+1.33e3T2 1 - 4T + 1.33e3T^{2}
13 1+54iT−2.19e3T2 1 + 54iT - 2.19e3T^{2}
17 1+14iT−4.91e3T2 1 + 14iT - 4.91e3T^{2}
19 1+92T+6.85e3T2 1 + 92T + 6.85e3T^{2}
23 1−152iT−1.21e4T2 1 - 152iT - 1.21e4T^{2}
29 1−106T+2.43e4T2 1 - 106T + 2.43e4T^{2}
31 1+144T+2.97e4T2 1 + 144T + 2.97e4T^{2}
37 1−158iT−5.06e4T2 1 - 158iT - 5.06e4T^{2}
41 1+390T+6.89e4T2 1 + 390T + 6.89e4T^{2}
43 1−508iT−7.95e4T2 1 - 508iT - 7.95e4T^{2}
47 1+528iT−1.03e5T2 1 + 528iT - 1.03e5T^{2}
53 1+606iT−1.48e5T2 1 + 606iT - 1.48e5T^{2}
59 1−364T+2.05e5T2 1 - 364T + 2.05e5T^{2}
61 1−678T+2.26e5T2 1 - 678T + 2.26e5T^{2}
67 1−844iT−3.00e5T2 1 - 844iT - 3.00e5T^{2}
71 1+8T+3.57e5T2 1 + 8T + 3.57e5T^{2}
73 1−422iT−3.89e5T2 1 - 422iT - 3.89e5T^{2}
79 1+384T+4.93e5T2 1 + 384T + 4.93e5T^{2}
83 1−548iT−5.71e5T2 1 - 548iT - 5.71e5T^{2}
89 1+1.19e3T+7.04e5T2 1 + 1.19e3T + 7.04e5T^{2}
97 1+1.50e3iT−9.12e5T2 1 + 1.50e3iT - 9.12e5T^{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

−8.542222737489697285052910572286, −8.245525450570863198102638050201, −7.23360785145816925042800614774, −6.70870919036987273729668193862, −5.71584198271652716078803442801, −5.03452139139164025086740393084, −3.85464774016382354694199052601, −3.03862808860091806089384428942, −1.91986023388424447428465672186, −0.893914474519227313754950317785, 0.20740427915651909944462589503, 1.78937730751274217184649329827, 2.64518240234891050233234077620, 3.86524136341126907758285643072, 4.45315358010991380576863299478, 5.34907356905472767499163831716, 6.32606465557654087179360601218, 6.86202624460057260219195230945, 8.014987727968908900509961073315, 8.873710292611585553740468772235

Graph of the ZZ-function along the critical line