↑

Properties

Label 2-3528-24.5-c0-0-3
Degree 22
Conductor 35283528
Sign −0.985−0.169i-0.985 - 0.169i
Analytic cond. 1.760701.76070
Root an. cond. 1.326911.32691
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s − 1.41·11-s − 1.00·16-s + (1.00 + 1.00i)22-s − 1.41i·23-s − 25-s − 1.41·29-s + (0.707 + 0.707i)32-s + 2i·37-s − 2i·43-s − 1.41i·44-s + (−1.00 + 1.00i)46-s + (0.707 + 0.707i)50-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s − 1.41·11-s − 1.00·16-s + (1.00 + 1.00i)22-s − 1.41i·23-s − 25-s − 1.41·29-s + (0.707 + 0.707i)32-s + 2i·37-s − 2i·43-s − 1.41i·44-s + (−1.00 + 1.00i)46-s + (0.707 + 0.707i)50-s + ⋯

Functional equation

Λ(s)=(3528s/2 ΓC(s) L(s)=((−0.985−0.169i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3528s/2 ΓC(s) L(s)=((−0.985−0.169i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35283528    =    23⋅32⋅722^{3} \cdot 3^{2} \cdot 7^{2}
Sign: −0.985−0.169i-0.985 - 0.169i
Analytic conductor: 1.760701.76070
Root analytic conductor: 1.326911.32691
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3528(197,⋅)\chi_{3528} (197, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3528, ( :0), −0.985−0.169i)(2,\ 3528,\ (\ :0),\ -0.985 - 0.169i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.19050530940.1905053094
L(12)L(\frac12) ≈\approx 0.19050530940.1905053094
L(1)L(1) 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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
3 1 1
7 1 1
good5 1+T2 1 + T^{2}
11 1+1.41T+T2 1 + 1.41T + T^{2}
13 1−T2 1 - T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+1.41iT−T2 1 + 1.41iT - T^{2}
29 1+1.41T+T2 1 + 1.41T + T^{2}
31 1+T2 1 + T^{2}
37 1−2iT−T2 1 - 2iT - T^{2}
41 1−T2 1 - T^{2}
43 1+2iT−T2 1 + 2iT - T^{2}
47 1−T2 1 - T^{2}
53 1+1.41T+T2 1 + 1.41T + T^{2}
59 1+T2 1 + T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1+1.41iT−T2 1 + 1.41iT - T^{2}
73 1+T2 1 + T^{2}
79 1+2T+T2 1 + 2T + T^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - T^{2}
97 1+T2 1 + T^{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

−8.378538841430262635017221725683, −7.82975588358122350536019560752, −7.15820660341272734156273639287, −6.21607743308674196319828175543, −5.20559925027765817120152208585, −4.39548234458146003576983714281, −3.41054478746214282874386287356, −2.59043182858815207006197770108, −1.75441314570331899574368621344, −0.13299040740965585101162682182, 1.57325743811055333423216118893, 2.55334368382664277380309793610, 3.78433654928380714870820229851, 4.86566617807112408505781935175, 5.62172937055521596787328557526, 6.03956018819755531990388225411, 7.28488622847550263371532109516, 7.60579720839669954640074633674, 8.214374287766958627495451252801, 9.184631271161157954234269190776

Graph of the ZZ-function along the critical line