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Properties

Label 2-2800-5.4-c1-0-23
Degree 22
Conductor 28002800
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 22.358122.3581
Root an. cond. 4.728434.72843
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  + 3i·3-s − i·7-s − 6·9-s + 5·11-s − 6i·13-s + i·17-s − 3·19-s + 3·21-s − 9i·27-s + 6·29-s + 4·31-s + 15i·33-s − 8i·37-s + 18·39-s + 11·41-s + ⋯
L(s)  = 1  + 1.73i·3-s − 0.377i·7-s − 2·9-s + 1.50·11-s − 1.66i·13-s + 0.242i·17-s − 0.688·19-s + 0.654·21-s − 1.73i·27-s + 1.11·29-s + 0.718·31-s + 2.61i·33-s − 1.31i·37-s + 2.88·39-s + 1.71·41-s + ⋯

Functional equation

Λ(s)=(2800s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2800s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{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: 28002800    =    24⋅52⋅72^{4} \cdot 5^{2} \cdot 7
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 22.358122.3581
Root analytic conductor: 4.728434.72843
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2800(449,⋅)\chi_{2800} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2800, ( :1/2), 0.447−0.894i)(2,\ 2800,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.9197624981.919762498
L(12)L(\frac12) ≈\approx 1.9197624981.919762498
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
5 1 1
7 1+iT 1 + iT
good3 1−3iT−3T2 1 - 3iT - 3T^{2}
11 1−5T+11T2 1 - 5T + 11T^{2}
13 1+6iT−13T2 1 + 6iT - 13T^{2}
17 1−iT−17T2 1 - iT - 17T^{2}
19 1+3T+19T2 1 + 3T + 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+8iT−37T2 1 + 8iT - 37T^{2}
41 1−11T+41T2 1 - 11T + 41T^{2}
43 1−8iT−43T2 1 - 8iT - 43T^{2}
47 1−2iT−47T2 1 - 2iT - 47T^{2}
53 1−4iT−53T2 1 - 4iT - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−9iT−67T2 1 - 9iT - 67T^{2}
71 1−10T+71T2 1 - 10T + 71T^{2}
73 1+7iT−73T2 1 + 7iT - 73T^{2}
79 1+2T+79T2 1 + 2T + 79T^{2}
83 1+11iT−83T2 1 + 11iT - 83T^{2}
89 1−11T+89T2 1 - 11T + 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

−9.076235760795153340601652790456, −8.416205533675640242505669875373, −7.60162871113490771013261133864, −6.35568638090845137880446584403, −5.81915723962884947796499165911, −4.80025853461378712326735180606, −4.19450755223027610189939265590, −3.53629591371083024295916370386, −2.66749320425673017600255589050, −0.849514495136806981317882134586, 0.930520821249209750120031638115, 1.81937509470877097482148290782, 2.54685890465382673269306460434, 3.83965706151057570278237292129, 4.77994703781332761587988288891, 6.04844473627330597333054081040, 6.60881873513603227130121832174, 6.85712782025260333711457713729, 7.85526139618735009564756144543, 8.680502964898596526145521334914

Graph of the ZZ-function along the critical line