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Properties

Label 2-30e2-5.4-c3-0-9
Degree 22
Conductor 900900
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 53.101753.1017
Root an. cond. 7.287097.28709
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  − 7i·7-s + 54·11-s + 55i·13-s − 18i·17-s + 25·19-s − 18i·23-s − 54·29-s − 271·31-s + 314i·37-s + 360·41-s + 163i·43-s − 522i·47-s + 294·49-s − 36i·53-s + 126·59-s + ⋯
L(s)  = 1  − 0.377i·7-s + 1.48·11-s + 1.17i·13-s − 0.256i·17-s + 0.301·19-s − 0.163i·23-s − 0.345·29-s − 1.57·31-s + 1.39i·37-s + 1.37·41-s + 0.578i·43-s − 1.62i·47-s + 0.857·49-s − 0.0933i·53-s + 0.278·59-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s+3/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 53.101753.1017
Root analytic conductor: 7.287097.28709
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ900(649,⋅)\chi_{900} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :3/2), 0.894−0.447i)(2,\ 900,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 2.1962411522.196241152
L(12)L(\frac12) ≈\approx 2.1962411522.196241152
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 1
5 1 1
good7 1+7iT−343T2 1 + 7iT - 343T^{2}
11 1−54T+1.33e3T2 1 - 54T + 1.33e3T^{2}
13 1−55iT−2.19e3T2 1 - 55iT - 2.19e3T^{2}
17 1+18iT−4.91e3T2 1 + 18iT - 4.91e3T^{2}
19 1−25T+6.85e3T2 1 - 25T + 6.85e3T^{2}
23 1+18iT−1.21e4T2 1 + 18iT - 1.21e4T^{2}
29 1+54T+2.43e4T2 1 + 54T + 2.43e4T^{2}
31 1+271T+2.97e4T2 1 + 271T + 2.97e4T^{2}
37 1−314iT−5.06e4T2 1 - 314iT - 5.06e4T^{2}
41 1−360T+6.89e4T2 1 - 360T + 6.89e4T^{2}
43 1−163iT−7.95e4T2 1 - 163iT - 7.95e4T^{2}
47 1+522iT−1.03e5T2 1 + 522iT - 1.03e5T^{2}
53 1+36iT−1.48e5T2 1 + 36iT - 1.48e5T^{2}
59 1−126T+2.05e5T2 1 - 126T + 2.05e5T^{2}
61 1−47T+2.26e5T2 1 - 47T + 2.26e5T^{2}
67 1+343iT−3.00e5T2 1 + 343iT - 3.00e5T^{2}
71 1−1.08e3T+3.57e5T2 1 - 1.08e3T + 3.57e5T^{2}
73 1−1.05e3iT−3.89e5T2 1 - 1.05e3iT - 3.89e5T^{2}
79 1−568T+4.93e5T2 1 - 568T + 4.93e5T^{2}
83 1−1.42e3iT−5.71e5T2 1 - 1.42e3iT - 5.71e5T^{2}
89 1−1.44e3T+7.04e5T2 1 - 1.44e3T + 7.04e5T^{2}
97 1+439iT−9.12e5T2 1 + 439iT - 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

−9.492121335054437903208358807526, −9.214571536136264920504379289044, −8.130800541329968481664752096672, −7.01019777164308631751311503509, −6.58937134875236234111564721511, −5.40316328573450888373305853268, −4.25460544231362368151426102862, −3.61310313425532991933151187177, −2.08055695867881172962423382344, −0.978311506715119554673637558918, 0.71826212018856001737328380193, 1.99150276417406948795029389390, 3.31631711913275901123032187752, 4.15261123177848659814572961117, 5.45849187789021591257034664522, 6.06772443958902467981983828011, 7.17845278542478162134890350893, 7.923206977798691821157488472972, 9.082140258860466943296266022024, 9.378385635052092194339047562209

Graph of the ZZ-function along the critical line