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Properties

Label 2-1200-5.4-c3-0-37
Degree 22
Conductor 12001200
Sign 0.447+0.894i0.447 + 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 − 5i·7-s − 9·9-s − 14·11-s − i·13-s + 46i·17-s + 19·19-s + 15·21-s − 46i·23-s − 27i·27-s − 14·29-s − 133·31-s − 42i·33-s + 258i·37-s + 3·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.269i·7-s − 0.333·9-s − 0.383·11-s − 0.0213i·13-s + 0.656i·17-s + 0.229·19-s + 0.155·21-s − 0.417i·23-s − 0.192i·27-s − 0.0896·29-s − 0.770·31-s − 0.221i·33-s + 1.14i·37-s + 0.0123·39-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+0.894i0.447 + 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 1.2456738521.245673852
L(12)L(\frac12) ≈\approx 1.2456738521.245673852
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+5iT−343T2 1 + 5iT - 343T^{2}
11 1+14T+1.33e3T2 1 + 14T + 1.33e3T^{2}
13 1+iT−2.19e3T2 1 + iT - 2.19e3T^{2}
17 1−46iT−4.91e3T2 1 - 46iT - 4.91e3T^{2}
19 1−19T+6.85e3T2 1 - 19T + 6.85e3T^{2}
23 1+46iT−1.21e4T2 1 + 46iT - 1.21e4T^{2}
29 1+14T+2.43e4T2 1 + 14T + 2.43e4T^{2}
31 1+133T+2.97e4T2 1 + 133T + 2.97e4T^{2}
37 1−258iT−5.06e4T2 1 - 258iT - 5.06e4T^{2}
41 1−84T+6.89e4T2 1 - 84T + 6.89e4T^{2}
43 1+167iT−7.95e4T2 1 + 167iT - 7.95e4T^{2}
47 1+410iT−1.03e5T2 1 + 410iT - 1.03e5T^{2}
53 1+456iT−1.48e5T2 1 + 456iT - 1.48e5T^{2}
59 1+194T+2.05e5T2 1 + 194T + 2.05e5T^{2}
61 1+17T+2.26e5T2 1 + 17T + 2.26e5T^{2}
67 1+653iT−3.00e5T2 1 + 653iT - 3.00e5T^{2}
71 1+828T+3.57e5T2 1 + 828T + 3.57e5T^{2}
73 1+570iT−3.89e5T2 1 + 570iT - 3.89e5T^{2}
79 1+552T+4.93e5T2 1 + 552T + 4.93e5T^{2}
83 1−142iT−5.71e5T2 1 - 142iT - 5.71e5T^{2}
89 1−1.10e3T+7.04e5T2 1 - 1.10e3T + 7.04e5T^{2}
97 1−841iT−9.12e5T2 1 - 841iT - 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

−9.236608865731083763355389776747, −8.490078953783317211010443185000, −7.66770248606964398152314325281, −6.72043598227828869950645212363, −5.75688026059346262893877801932, −4.92064878766796327484429902369, −3.98829699584281758309677404845, −3.10458170886009886005081542092, −1.84938106257661067139180973486, −0.33226466449880231217591547039, 0.997453533672714072584271601255, 2.22803927767488163653189640446, 3.13964558403526928726489327418, 4.37812333010377834639422300518, 5.46025372065920046688909614585, 6.09493377238305412731079992222, 7.28703502884847903674895974005, 7.63425352151050579720158848178, 8.777134374669988223617351143887, 9.345351384744648606772442839705

Graph of the ZZ-function along the critical line