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Properties

Label 2-1200-5.4-c3-0-16
Degree 22
Conductor 12001200
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 70.802270.8022
Root an. cond. 8.414408.41440
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 7i·7-s − 9·9-s + 54·11-s + 55i·13-s + 18i·17-s − 25·19-s − 21·21-s − 18i·23-s − 27i·27-s + 54·29-s + 271·31-s + 162i·33-s + 314i·37-s − 165·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 1.48·11-s + 1.17i·13-s + 0.256i·17-s − 0.301·19-s − 0.218·21-s − 0.163i·23-s − 0.192i·27-s + 0.345·29-s + 1.57·31-s + 0.854i·33-s + 1.39i·37-s − 0.677·39-s + ⋯

Functional equation

Λ(s)=(1200s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1200s/2 ΓC(s+3/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12001200    =    24⋅3⋅522^{4} \cdot 3 \cdot 5^{2}
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 70.802270.8022
Root analytic conductor: 8.414408.41440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1200(49,⋅)\chi_{1200} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1200, ( :3/2), −0.447−0.894i)(2,\ 1200,\ (\ :3/2),\ -0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 2.0173763152.017376315
L(12)L(\frac12) ≈\approx 2.0173763152.017376315
L(52)L(\frac{5}{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 1−3iT 1 - 3iT
5 1 1
good7 1−7iT−343T2 1 - 7iT - 343T^{2}
11 1−54T+1.33e3T2 1 - 54T + 1.33e3T^{2}
13 1−55iT−2.19e3T2 1 - 55iT - 2.19e3T^{2}
17 1−18iT−4.91e3T2 1 - 18iT - 4.91e3T^{2}
19 1+25T+6.85e3T2 1 + 25T + 6.85e3T^{2}
23 1+18iT−1.21e4T2 1 + 18iT - 1.21e4T^{2}
29 1−54T+2.43e4T2 1 - 54T + 2.43e4T^{2}
31 1−271T+2.97e4T2 1 - 271T + 2.97e4T^{2}
37 1−314iT−5.06e4T2 1 - 314iT - 5.06e4T^{2}
41 1+360T+6.89e4T2 1 + 360T + 6.89e4T^{2}
43 1+163iT−7.95e4T2 1 + 163iT - 7.95e4T^{2}
47 1+522iT−1.03e5T2 1 + 522iT - 1.03e5T^{2}
53 1−36iT−1.48e5T2 1 - 36iT - 1.48e5T^{2}
59 1−126T+2.05e5T2 1 - 126T + 2.05e5T^{2}
61 1−47T+2.26e5T2 1 - 47T + 2.26e5T^{2}
67 1−343iT−3.00e5T2 1 - 343iT - 3.00e5T^{2}
71 1−1.08e3T+3.57e5T2 1 - 1.08e3T + 3.57e5T^{2}
73 1−1.05e3iT−3.89e5T2 1 - 1.05e3iT - 3.89e5T^{2}
79 1+568T+4.93e5T2 1 + 568T + 4.93e5T^{2}
83 1−1.42e3iT−5.71e5T2 1 - 1.42e3iT - 5.71e5T^{2}
89 1+1.44e3T+7.04e5T2 1 + 1.44e3T + 7.04e5T^{2}
97 1+439iT−9.12e5T2 1 + 439iT - 9.12e5T^{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

−9.704153871904488121429806842047, −8.721794412025526577769542282224, −8.424326200094274154299585098569, −6.81318786416026731162709429189, −6.51104638832887127008378911074, −5.32049161632393736882652333550, −4.34482991094341921657101712414, −3.70631398055409540235275342180, −2.40772034300162293554848723998, −1.21639446469872368141758830268, 0.52899772828367240462423674364, 1.44944424887702306612152415128, 2.78148908002658107576742104953, 3.78049889278053770911545713004, 4.79937358438954831285970096414, 5.95482858596549819677102579659, 6.59644822228975531186182986916, 7.45582428244387797958789774062, 8.227156027655877752073021621797, 9.043489549090572496222499366155

Graph of the ZZ-function along the critical line