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Properties

Label 2-5400-5.4-c1-0-25
Degree 22
Conductor 54005400
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 43.119243.1192
Root an. cond. 6.566526.56652
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  + 2i·7-s + 11-s − i·13-s − i·17-s − 4·19-s − i·23-s + 5·29-s + 31-s + 6i·37-s − 7i·43-s + 7i·47-s + 3·49-s + 12i·53-s + 4·59-s + 10·61-s + ⋯
L(s)  = 1  + 0.755i·7-s + 0.301·11-s − 0.277i·13-s − 0.242i·17-s − 0.917·19-s − 0.208i·23-s + 0.928·29-s + 0.179·31-s + 0.986i·37-s − 1.06i·43-s + 1.02i·47-s + 0.428·49-s + 1.64i·53-s + 0.520·59-s + 1.28·61-s + ⋯

Functional equation

Λ(s)=(5400s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5400s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 54005400    =    23⋅33⋅522^{3} \cdot 3^{3} \cdot 5^{2}
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 43.119243.1192
Root analytic conductor: 6.566526.56652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5400(649,⋅)\chi_{5400} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5400, ( :1/2), 0.447−0.894i)(2,\ 5400,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.7099751221.709975122
L(12)L(\frac12) ≈\approx 1.7099751221.709975122
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 1
5 1 1
good7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1−T+11T2 1 - T + 11T^{2}
13 1+iT−13T2 1 + iT - 13T^{2}
17 1+iT−17T2 1 + iT - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+iT−23T2 1 + iT - 23T^{2}
29 1−5T+29T2 1 - 5T + 29T^{2}
31 1−T+31T2 1 - T + 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+7iT−43T2 1 + 7iT - 43T^{2}
47 1−7iT−47T2 1 - 7iT - 47T^{2}
53 1−12iT−53T2 1 - 12iT - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1+4iT−67T2 1 + 4iT - 67T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1+15T+79T2 1 + 15T + 79T^{2}
83 1+2iT−83T2 1 + 2iT - 83T^{2}
89 1−12T+89T2 1 - 12T + 89T^{2}
97 1−10iT−97T2 1 - 10iT - 97T^{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

−8.385505387719458446938286564844, −7.67070689557175807334461262451, −6.72779144001320854807441256832, −6.22505549548480915382681375905, −5.41465136428094452442428538847, −4.68230215194648102845723579985, −3.86943734200286838567029651915, −2.85643756423026950406725372067, −2.19035269333625169688436513937, −0.961687572878066953455925939998, 0.53453983323481518663677549440, 1.66932950422626434459552698595, 2.63368115076741450350968166267, 3.75833817454951547784460863793, 4.20806667609395150143893487289, 5.09962748974859434969231042613, 5.95743199083639168433177707665, 6.78995086703178448939484440777, 7.13515111669526287765585740295, 8.224321525524111180145217636878

Graph of the ZZ-function along the critical line