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Properties

Label 2-2550-5.4-c1-0-31
Degree 22
Conductor 25502550
Sign 0.894+0.447i0.894 + 0.447i
Analytic cond. 20.361820.3618
Root an. cond. 4.512414.51241
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·2-s − i·3-s − 4-s + 6-s − i·8-s − 9-s + 4·11-s + i·12-s − 2i·13-s + 16-s + i·17-s − i·18-s + 4·19-s + 4i·22-s − 4i·23-s − 24-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s − 0.554i·13-s + 0.250·16-s + 0.242i·17-s − 0.235i·18-s + 0.917·19-s + 0.852i·22-s − 0.834i·23-s − 0.204·24-s + ⋯

Functional equation

Λ(s)=(2550s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2550s/2 ΓC(s+1/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{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: 25502550    =    2⋅3⋅52⋅172 \cdot 3 \cdot 5^{2} \cdot 17
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 20.361820.3618
Root analytic conductor: 4.512414.51241
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2550(2449,⋅)\chi_{2550} (2449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2550, ( :1/2), 0.894+0.447i)(2,\ 2550,\ (\ :1/2),\ 0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.6828612871.682861287
L(12)L(\frac12) ≈\approx 1.6828612871.682861287
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−iT 1 - iT
3 1+iT 1 + iT
5 1 1
17 1−iT 1 - iT
good7 1−7T2 1 - 7T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+6iT−37T2 1 + 6iT - 37T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1−8iT−43T2 1 - 8iT - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1+6iT−53T2 1 + 6iT - 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1+8iT−67T2 1 + 8iT - 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1−2iT−73T2 1 - 2iT - 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1−14T+89T2 1 - 14T + 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

−8.674976872205014579610399513176, −8.035609656333998927155692578483, −7.17973435817003348097957125423, −6.67711219316901086933526183224, −5.84623157877719780983481316184, −5.15512757928069106842408712785, −4.05414912395753202368217812084, −3.23934968937428611353574434117, −1.86874871860044182921554026911, −0.66659293232709081740859685894, 1.11452878732348489724099533534, 2.20686752998281625449359413655, 3.49275402817289348986607960888, 3.86206364322777521341628953009, 4.93038821072975749988228583327, 5.59984040164011358027007334065, 6.68157484600785741437958656523, 7.42637657357619837608742266578, 8.561070558430045159539506325686, 9.102686175362718376924115419268

Graph of the ZZ-function along the critical line