↑

Properties

Label 2-1100-5.4-c5-0-8
Degree 22
Conductor 11001100
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 176.422176.422
Root an. cond. 13.282413.2824
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.57i·3-s + 209. i·7-s + 222.·9-s − 121·11-s + 100. i·13-s + 978. i·17-s − 1.35e3·19-s − 957.·21-s + 2.07e3i·23-s + 2.12e3i·27-s + 4.87e3·29-s − 6.30e3·31-s − 553. i·33-s − 541. i·37-s − 460.·39-s + ⋯
L(s)  = 1  + 0.293i·3-s + 1.61i·7-s + 0.913·9-s − 0.301·11-s + 0.165i·13-s + 0.821i·17-s − 0.859·19-s − 0.474·21-s + 0.818i·23-s + 0.562i·27-s + 1.07·29-s − 1.17·31-s − 0.0885i·33-s − 0.0650i·37-s − 0.0484·39-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+5/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 176.422176.422
Root analytic conductor: 13.282413.2824
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ1100(749,⋅)\chi_{1100} (749, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :5/2), −0.894+0.447i)(2,\ 1100,\ (\ :5/2),\ -0.894 + 0.447i)

Particular Values

L(3)L(3) ≈\approx 1.1165889541.116588954
L(12)L(\frac12) ≈\approx 1.1165889541.116588954
L(72)L(\frac{7}{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
11 1+121T 1 + 121T
good3 1−4.57iT−243T2 1 - 4.57iT - 243T^{2}
7 1−209.iT−1.68e4T2 1 - 209. iT - 1.68e4T^{2}
13 1−100.iT−3.71e5T2 1 - 100. iT - 3.71e5T^{2}
17 1−978.iT−1.41e6T2 1 - 978. iT - 1.41e6T^{2}
19 1+1.35e3T+2.47e6T2 1 + 1.35e3T + 2.47e6T^{2}
23 1−2.07e3iT−6.43e6T2 1 - 2.07e3iT - 6.43e6T^{2}
29 1−4.87e3T+2.05e7T2 1 - 4.87e3T + 2.05e7T^{2}
31 1+6.30e3T+2.86e7T2 1 + 6.30e3T + 2.86e7T^{2}
37 1+541.iT−6.93e7T2 1 + 541. iT - 6.93e7T^{2}
41 1+1.11e4T+1.15e8T2 1 + 1.11e4T + 1.15e8T^{2}
43 1+9.10e3iT−1.47e8T2 1 + 9.10e3iT - 1.47e8T^{2}
47 1−2.78e4iT−2.29e8T2 1 - 2.78e4iT - 2.29e8T^{2}
53 1+7.75e3iT−4.18e8T2 1 + 7.75e3iT - 4.18e8T^{2}
59 1−3.83e4T+7.14e8T2 1 - 3.83e4T + 7.14e8T^{2}
61 1+4.35e4T+8.44e8T2 1 + 4.35e4T + 8.44e8T^{2}
67 1−1.65e3iT−1.35e9T2 1 - 1.65e3iT - 1.35e9T^{2}
71 1−2.69e4T+1.80e9T2 1 - 2.69e4T + 1.80e9T^{2}
73 1−8.28e4iT−2.07e9T2 1 - 8.28e4iT - 2.07e9T^{2}
79 1+6.33e3T+3.07e9T2 1 + 6.33e3T + 3.07e9T^{2}
83 1+5.17e4iT−3.93e9T2 1 + 5.17e4iT - 3.93e9T^{2}
89 1−1.41e4T+5.58e9T2 1 - 1.41e4T + 5.58e9T^{2}
97 1+8.30e4iT−8.58e9T2 1 + 8.30e4iT - 8.58e9T^{2}
show more
show less
   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

−9.541177384378772181190053386849, −8.808493736649738691112404048374, −8.145843133349777116901772857597, −7.06373372430142317374029304125, −6.13299368490930207642538136159, −5.38228253229574639388389998816, −4.48475517861725488588317987357, −3.45371093492035312334919108288, −2.32028000850917794412946737330, −1.51109798408930275191290736642, 0.21958748931528972100012494695, 1.00399527498375262880513036906, 2.10655300103796123117836750873, 3.42500557979857258995416102105, 4.31498130189266359209343360472, 4.99899325181694724794029351008, 6.45008530668366185101137008872, 7.01382528753420104195893889373, 7.66666875332713262056129542470, 8.520249458058456533699302828888

Graph of the ZZ-function along the critical line