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Properties

Label 2-7200-5.4-c1-0-57
Degree 22
Conductor 72007200
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 57.492257.4922
Root an. cond. 7.582367.58236
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 − 5·11-s + 5i·17-s − 5·19-s − 6i·23-s + 4·29-s + 10·31-s + 10i·37-s − 5·41-s + 4i·43-s − 8i·47-s + 3·49-s + 10i·53-s − 10·61-s − 3i·67-s + ⋯
L(s)  = 1  + 0.755i·7-s − 1.50·11-s + 1.21i·17-s − 1.14·19-s − 1.25i·23-s + 0.742·29-s + 1.79·31-s + 1.64i·37-s − 0.780·41-s + 0.609i·43-s − 1.16i·47-s + 0.428·49-s + 1.37i·53-s − 1.28·61-s − 0.366i·67-s + ⋯

Functional equation

Λ(s)=(7200s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(7200s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{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: 72007200    =    25⋅32⋅522^{5} \cdot 3^{2} \cdot 5^{2}
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 57.492257.4922
Root analytic conductor: 7.582367.58236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ7200(6049,⋅)\chi_{7200} (6049, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 7200, ( :1/2), −0.447+0.894i)(2,\ 7200,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.35889275830.3588927583
L(12)L(\frac12) ≈\approx 0.35889275830.3588927583
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+5T+11T2 1 + 5T + 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1−5iT−17T2 1 - 5iT - 17T^{2}
19 1+5T+19T2 1 + 5T + 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1−10T+31T2 1 - 10T + 31T^{2}
37 1−10iT−37T2 1 - 10iT - 37T^{2}
41 1+5T+41T2 1 + 5T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1+8iT−47T2 1 + 8iT - 47T^{2}
53 1−10iT−53T2 1 - 10iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 1+3iT−67T2 1 + 3iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+5iT−73T2 1 + 5iT - 73T^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1−iT−83T2 1 - iT - 83T^{2}
89 1+9T+89T2 1 + 9T + 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

−8.001200429110217781127095934640, −6.86321698490405171872989459535, −6.25539741310602627489639645224, −5.69716245654546124102327817574, −4.73195858614391250243416121206, −4.35102195707219607805513246131, −2.97590832759699342276580701877, −2.60039703955394184412487515662, −1.58237202439360885511901440437, −0.096609520968591597397577249996, 0.946771510326957582099687517257, 2.26004773416192368350231326520, 2.89849467608506502650243031881, 3.85406954554083744119136396546, 4.67439936844427499058545876276, 5.22633083961688524811836886884, 6.05001814601367796966109426352, 6.89227647999716666087434851707, 7.48120076200818950937611115087, 8.025598430871014784734312261274

Graph of the ZZ-function along the critical line