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Properties

Label 2-4800-5.4-c1-0-25
Degree 22
Conductor 48004800
Sign 0.894+0.447i0.894 + 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 − 9-s − 4·11-s − 2i·13-s + 2i·17-s − 4·19-s + 8i·23-s + i·27-s + 6·29-s + 8·31-s + 4i·33-s − 6i·37-s − 2·39-s − 6·41-s + 4i·43-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.333·9-s − 1.20·11-s − 0.554i·13-s + 0.485i·17-s − 0.917·19-s + 1.66i·23-s + 0.192i·27-s + 1.11·29-s + 1.43·31-s + 0.696i·33-s − 0.986i·37-s − 0.320·39-s − 0.937·41-s + 0.609i·43-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+0.447i0.894 + 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 1.5077809541.507780954
L(12)L(\frac12) ≈\approx 1.5077809541.507780954
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 1−7T2 1 - 7T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1+2iT−13T2 1 + 2iT - 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 1−8T+31T2 1 - 8T + 31T^{2}
37 1+6iT−37T2 1 + 6iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1+2iT−53T2 1 + 2iT - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1+10iT−73T2 1 + 10iT - 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1−6T+89T2 1 - 6T + 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

−8.130590170972972603076107966712, −7.63051559102022815183868598274, −6.81176471181065967300830837226, −6.04729437280323907624897614749, −5.39710295019085920571810835142, −4.62616476056665332402059815050, −3.54897778581695383335689555321, −2.72164673517412790442608165119, −1.89406437565412581366063044597, −0.66426287377050287659223418108, 0.65232441773682362223888611681, 2.29913613008558667759988059382, 2.80171671556360785553589144335, 3.93918833559908021470805200419, 4.74447841856728311489763484640, 5.12416914069215578975935401843, 6.31277083125894241879992294326, 6.69108807978462127774149363164, 7.76738980779181533128072151681, 8.482869050102263211313531019054

Graph of the ZZ-function along the critical line