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Properties

Label 2-2400-5.4-c1-0-8
Degree 22
Conductor 24002400
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 19.164019.1640
Root an. cond. 4.377684.37768
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 − 8·19-s + 4i·23-s − i·27-s + 6·29-s + 4i·33-s − 2i·37-s − 2·39-s − 6·41-s + 4i·43-s + 12i·47-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.333·9-s + 1.20·11-s + 0.554i·13-s + 0.485i·17-s − 1.83·19-s + 0.834i·23-s − 0.192i·27-s + 1.11·29-s + 0.696i·33-s − 0.328i·37-s − 0.320·39-s − 0.937·41-s + 0.609i·43-s + 1.75i·47-s + ⋯

Functional equation

Λ(s)=(2400s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2400s/2 ΓC(s+1/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{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: 24002400    =    25⋅3⋅522^{5} \cdot 3 \cdot 5^{2}
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 19.164019.1640
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2400(1249,⋅)\chi_{2400} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2400, ( :1/2), −0.447−0.894i)(2,\ 2400,\ (\ :1/2),\ -0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.4220154971.422015497
L(12)L(\frac12) ≈\approx 1.4220154971.422015497
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+8T+19T2 1 + 8T + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1−12iT−47T2 1 - 12iT - 47T^{2}
53 1+6iT−53T2 1 + 6iT - 53T^{2}
59 1+12T+59T2 1 + 12T + 59T^{2}
61 1−14T+61T2 1 - 14T + 61T^{2}
67 1−12iT−67T2 1 - 12iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−2iT−73T2 1 - 2iT - 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1+2T+89T2 1 + 2T + 89T^{2}
97 1−14iT−97T2 1 - 14iT - 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

−9.121279321727361796187495882666, −8.653946391450262269850011610284, −7.80110789286529074190981222881, −6.59253762829136573373045722505, −6.33179208332781865686470221518, −5.17977672209596164305187515504, −4.22025835593867010029583925639, −3.80609050352547795382138013611, −2.52612844324430918158692015706, −1.37731319719758585072401552200, 0.49200684404884067164167360943, 1.76875518447980779067063967005, 2.74778045478718415433800835631, 3.85858199368142659427749753835, 4.68200779026234768406191851070, 5.72318174511009985932257059866, 6.66034590661104915103867788837, 6.86163984448130820996516660373, 8.132350852563471362107334593035, 8.566199376355159919024556049375

Graph of the ZZ-function along the critical line