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Properties

Label 2-7200-40.29-c1-0-48
Degree 22
Conductor 72007200
Sign 0.979+0.200i0.979 + 0.200i
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  − 0.732i·7-s + 2i·11-s + 3.46·13-s − 3.46i·17-s − 0.535i·19-s + 6.19i·23-s − 6.92i·29-s + 5.46·31-s − 2·37-s − 1.46·41-s − 5.26·43-s + 3.26i·47-s + 6.46·49-s + 11.4·53-s − 7.46i·59-s + ⋯
L(s)  = 1  − 0.276i·7-s + 0.603i·11-s + 0.960·13-s − 0.840i·17-s − 0.122i·19-s + 1.29i·23-s − 1.28i·29-s + 0.981·31-s − 0.328·37-s − 0.228·41-s − 0.803·43-s + 0.476i·47-s + 0.923·49-s + 1.57·53-s − 0.971i·59-s + ⋯

Functional equation

Λ(s)=(7200s/2 ΓC(s) L(s)=((0.979+0.200i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(7200s/2 ΓC(s+1/2) L(s)=((0.979+0.200i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72007200    =    25⋅32⋅522^{5} \cdot 3^{2} \cdot 5^{2}
Sign: 0.979+0.200i0.979 + 0.200i
Analytic conductor: 57.492257.4922
Root analytic conductor: 7.582367.58236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ7200(2449,⋅)\chi_{7200} (2449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 7200, ( :1/2), 0.979+0.200i)(2,\ 7200,\ (\ :1/2),\ 0.979 + 0.200i)

Particular Values

L(1)L(1) ≈\approx 2.0842652642.084265264
L(12)L(\frac12) ≈\approx 2.0842652642.084265264
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+0.732iT−7T2 1 + 0.732iT - 7T^{2}
11 1−2iT−11T2 1 - 2iT - 11T^{2}
13 1−3.46T+13T2 1 - 3.46T + 13T^{2}
17 1+3.46iT−17T2 1 + 3.46iT - 17T^{2}
19 1+0.535iT−19T2 1 + 0.535iT - 19T^{2}
23 1−6.19iT−23T2 1 - 6.19iT - 23T^{2}
29 1+6.92iT−29T2 1 + 6.92iT - 29T^{2}
31 1−5.46T+31T2 1 - 5.46T + 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+1.46T+41T2 1 + 1.46T + 41T^{2}
43 1+5.26T+43T2 1 + 5.26T + 43T^{2}
47 1−3.26iT−47T2 1 - 3.26iT - 47T^{2}
53 1−11.4T+53T2 1 - 11.4T + 53T^{2}
59 1+7.46iT−59T2 1 + 7.46iT - 59T^{2}
61 1−8.92iT−61T2 1 - 8.92iT - 61T^{2}
67 1+10.7T+67T2 1 + 10.7T + 67T^{2}
71 1−5.46T+71T2 1 - 5.46T + 71T^{2}
73 1+7.46iT−73T2 1 + 7.46iT - 73T^{2}
79 1+1.07T+79T2 1 + 1.07T + 79T^{2}
83 1+1.26T+83T2 1 + 1.26T + 83T^{2}
89 1−8.92T+89T2 1 - 8.92T + 89T^{2}
97 1+14.3iT−97T2 1 + 14.3iT - 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

−7.78175824333411479741029275567, −7.28620291448221916503027154258, −6.52385434199599860394205420664, −5.83161830256803883609997193149, −5.05800875851543668427791665991, −4.28865933989080338133087181301, −3.59054252411118747913977732990, −2.70279185310123182703114814394, −1.72617063808679140399505043829, −0.71234773353581599026575818348, 0.789410375204736223278888177690, 1.78852375659628581147237991058, 2.81129965797299131306630869356, 3.57865029997606879931589363229, 4.29611518287951733345620207672, 5.20958251584781267163873218890, 5.90933823841472878331716956310, 6.49246977513353498071985751252, 7.14560743085105503330310366002, 8.220533277638198936551303026750

Graph of the ZZ-function along the critical line