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Properties

Label 2-6300-5.4-c1-0-30
Degree 22
Conductor 63006300
Sign 0.894+0.447i0.894 + 0.447i
Analytic cond. 50.305750.3057
Root an. cond. 7.092657.09265
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·7-s + 6·11-s − 2i·13-s + 4·19-s − 6i·23-s + 6·29-s + 8·31-s + 2i·37-s − 12·41-s + 4i·43-s − 12i·47-s − 49-s − 6i·53-s − 10·61-s + 8i·67-s + ⋯
L(s)  = 1  + 0.377i·7-s + 1.80·11-s − 0.554i·13-s + 0.917·19-s − 1.25i·23-s + 1.11·29-s + 1.43·31-s + 0.328i·37-s − 1.87·41-s + 0.609i·43-s − 1.75i·47-s − 0.142·49-s − 0.824i·53-s − 1.28·61-s + 0.977i·67-s + ⋯

Functional equation

Λ(s)=(6300s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(6300s/2 ΓC(s+1/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{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: 63006300    =    22⋅32⋅52⋅72^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 50.305750.3057
Root analytic conductor: 7.092657.09265
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ6300(6049,⋅)\chi_{6300} (6049, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 6300, ( :1/2), 0.894+0.447i)(2,\ 6300,\ (\ :1/2),\ 0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 2.3993721772.399372177
L(12)L(\frac12) ≈\approx 2.3993721772.399372177
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 1
5 1 1
7 1−iT 1 - iT
good11 1−6T+11T2 1 - 6T + 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1+12T+41T2 1 + 12T + 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+59T2 1 + 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 1−8iT−67T2 1 - 8iT - 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1−10iT−73T2 1 - 10iT - 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1+12iT−83T2 1 + 12iT - 83T^{2}
89 1−12T+89T2 1 - 12T + 89T^{2}
97 1+10iT−97T2 1 + 10iT - 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

−8.248811662233940481852217313048, −7.09202293056876309228527824723, −6.58274237140609938530377426043, −6.01516525757656425775262953203, −5.03256300300425476626843118864, −4.42735300891593009444764014884, −3.48434347903952138590673767717, −2.81514690297851522087089700088, −1.66048934383948113939919306771, −0.74438357632839803568015666113, 1.03261491511701919146318739714, 1.65603105868881597132470199974, 2.99850710010784147513581177145, 3.69508927659995564259377443052, 4.43040979552528500862834811718, 5.12358101024344324129832713009, 6.27549974579458084361128064107, 6.52017477764552950444850539455, 7.38827832652121158513470294345, 7.990084037998074328060509500241

Graph of the ZZ-function along the critical line