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Properties

Label 2-3528-392.125-c0-0-0
Degree 22
Conductor 35283528
Sign 0.926−0.375i0.926 - 0.375i
Analytic cond. 1.760701.76070
Root an. cond. 1.326911.32691
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.193 − 0.846i)5-s + (−0.222 − 0.974i)7-s + (−0.974 − 0.222i)8-s + (0.846 − 0.193i)10-s + (0.541 + 1.12i)11-s + (0.781 − 0.623i)14-s + (−0.222 − 0.974i)16-s + (0.541 + 0.678i)20-s + (−0.777 + 0.974i)22-s + (0.222 + 0.107i)25-s + (0.900 + 0.433i)28-s + (1.40 − 1.12i)29-s − 1.94i·31-s + (0.781 − 0.623i)32-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.193 − 0.846i)5-s + (−0.222 − 0.974i)7-s + (−0.974 − 0.222i)8-s + (0.846 − 0.193i)10-s + (0.541 + 1.12i)11-s + (0.781 − 0.623i)14-s + (−0.222 − 0.974i)16-s + (0.541 + 0.678i)20-s + (−0.777 + 0.974i)22-s + (0.222 + 0.107i)25-s + (0.900 + 0.433i)28-s + (1.40 − 1.12i)29-s − 1.94i·31-s + (0.781 − 0.623i)32-s + ⋯

Functional equation

Λ(s)=(3528s/2 ΓC(s) L(s)=((0.926−0.375i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3528s/2 ΓC(s) L(s)=((0.926−0.375i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35283528    =    23⋅32⋅722^{3} \cdot 3^{2} \cdot 7^{2}
Sign: 0.926−0.375i0.926 - 0.375i
Analytic conductor: 1.760701.76070
Root analytic conductor: 1.326911.32691
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3528(1693,⋅)\chi_{3528} (1693, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3528, ( :0), 0.926−0.375i)(2,\ 3528,\ (\ :0),\ 0.926 - 0.375i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5027293321.502729332
L(12)L(\frac12) ≈\approx 1.5027293321.502729332
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+(−0.433−0.900i)T 1 + (-0.433 - 0.900i)T
3 1 1
7 1+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
good5 1+(−0.193+0.846i)T+(−0.900−0.433i)T2 1 + (-0.193 + 0.846i)T + (-0.900 - 0.433i)T^{2}
11 1+(−0.541−1.12i)T+(−0.623+0.781i)T2 1 + (-0.541 - 1.12i)T + (-0.623 + 0.781i)T^{2}
13 1+(0.623−0.781i)T2 1 + (0.623 - 0.781i)T^{2}
17 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
19 1+T2 1 + T^{2}
23 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
29 1+(−1.40+1.12i)T+(0.222−0.974i)T2 1 + (-1.40 + 1.12i)T + (0.222 - 0.974i)T^{2}
31 1+1.94iT−T2 1 + 1.94iT - T^{2}
37 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
41 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
43 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
47 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
53 1+(−1.56−1.24i)T+(0.222+0.974i)T2 1 + (-1.56 - 1.24i)T + (0.222 + 0.974i)T^{2}
59 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
61 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
67 1−T2 1 - T^{2}
71 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
73 1+(−0.846+1.75i)T+(−0.623−0.781i)T2 1 + (-0.846 + 1.75i)T + (-0.623 - 0.781i)T^{2}
79 1−0.445T+T2 1 - 0.445T + T^{2}
83 1+(−0.781−0.376i)T+(0.623+0.781i)T2 1 + (-0.781 - 0.376i)T + (0.623 + 0.781i)T^{2}
89 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
97 1+1.56iT−T2 1 + 1.56iT - 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

−8.665299418752631944434864300281, −7.86854431581737041682165029058, −7.31307631696430860441137963471, −6.55032241723677504307022348627, −5.89852667128194701797790139800, −4.84085136285845607421408380143, −4.39250357269709809794190844371, −3.73284691530471343820472450027, −2.40622912982353829128922000849, −0.905173479951639148071386051799, 1.23990138328281876009946041289, 2.48287025638223140086078280845, 3.07888894101403832649915186550, 3.72709184483644822331654919368, 4.98076867672780404985658289451, 5.54499249989146292641485466570, 6.48604826945366230895502316808, 6.79529359804729337948407726883, 8.456463592717712572055569693144, 8.668748480650094934772638393798

Graph of the ZZ-function along the critical line