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Properties

Label 2-4800-5.4-c1-0-8
Degree 22
Conductor 48004800
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 38.328138.3281
Root an. cond. 6.190976.19097
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  + i·3-s + 4i·7-s − 9-s + 6i·13-s + 2i·17-s − 4·19-s − 4·21-s + 8i·23-s − i·27-s − 6·29-s − 6i·37-s − 6·39-s + 10·41-s − 4i·43-s + 8i·47-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.51i·7-s − 0.333·9-s + 1.66i·13-s + 0.485i·17-s − 0.917·19-s − 0.872·21-s + 1.66i·23-s − 0.192i·27-s − 1.11·29-s − 0.986i·37-s − 0.960·39-s + 1.56·41-s − 0.609i·43-s + 1.16i·47-s + ⋯

Functional equation

Λ(s)=(4800s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4800s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48004800    =    26⋅3⋅522^{6} \cdot 3 \cdot 5^{2}
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 38.328138.3281
Root analytic conductor: 6.190976.19097
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4800(3649,⋅)\chi_{4800} (3649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4800, ( :1/2), −0.894+0.447i)(2,\ 4800,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.1134814841.113481484
L(12)L(\frac12) ≈\approx 1.1134814841.113481484
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−iT 1 - iT
5 1 1
good7 1−4iT−7T2 1 - 4iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−6iT−13T2 1 - 6iT - 13T^{2}
17 1−2iT−17T2 1 - 2iT - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1−8iT−23T2 1 - 8iT - 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+6iT−37T2 1 + 6iT - 37T^{2}
41 1−10T+41T2 1 - 10T + 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+6T+61T2 1 + 6T + 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+14iT−73T2 1 + 14iT - 73T^{2}
79 1−16T+79T2 1 - 16T + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1+2T+89T2 1 + 2T + 89T^{2}
97 1+2iT−97T2 1 + 2iT - 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

−9.027889241861959885587486173493, −8.095145369001344704414318416768, −7.28500933842082239871001488842, −6.28633067041956117914708403611, −5.81690269783928820787763513248, −5.06493257573387129424754687241, −4.18252272051402475855765849818, −3.52216754910067126398048397586, −2.33005479422131672625401914617, −1.79099712360027512581802621209, 0.32704372559597997359209553918, 1.05011336400021207754614972501, 2.36712647095184738611668133738, 3.20983558407158514714468589104, 4.12770966459033456279726101144, 4.83230405348179457841508999607, 5.82105543763721700221125087307, 6.51288879626561545254674699794, 7.23130589566269958221591522735, 7.81612587206599255781830128993

Graph of the ZZ-function along the critical line