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Properties

Label 2-2550-5.4-c1-0-28
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 − 2i·7-s − i·8-s − 9-s − i·12-s + 2i·13-s + 2·14-s + 16-s + i·17-s − i·18-s + 4·19-s + 2·21-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 0.288i·12-s + 0.554i·13-s + 0.534·14-s + 0.250·16-s + 0.242i·17-s − 0.235i·18-s + 0.917·19-s + 0.436·21-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 1.5104771921.510477192
L(12)L(\frac12) ≈\approx 1.5104771921.510477192
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+2iT−7T2 1 + 2iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1+10T+31T2 1 + 10T + 31T^{2}
37 1+8iT−37T2 1 + 8iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 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 1−12T+59T2 1 - 12T + 59T^{2}
61 1−8T+61T2 1 - 8T + 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1−6T+71T2 1 - 6T + 71T^{2}
73 1−2iT−73T2 1 - 2iT - 73T^{2}
79 1−10T+79T2 1 - 10T + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1−18T+89T2 1 - 18T + 89T^{2}
97 1+14iT−97T2 1 + 14iT - 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

−8.972896758696052883051112482108, −8.193278888294562177613671787076, −7.26876052801828573588868265475, −6.84540714943143656177757857363, −5.73925305197396526310405932956, −5.14965293789467323514001022207, −4.08138226033058356006880172201, −3.70871785708933881972579842798, −2.25220788030208594698013464688, −0.62674347670875099260631967967, 0.991680808067481084250114185412, 2.05629181597688770640909017754, 2.97257568923183612670383180011, 3.72978536930896817466808366790, 5.09749030201889466224237422497, 5.54119792924237596299815788548, 6.48479365108209866322355732707, 7.56872729140441273204203570227, 7.993753500361472324876948423776, 9.095932902205181179578372046337

Graph of the ZZ-function along the critical line