↑

Properties

Label 2-3528-72.43-c0-0-5
Degree 22
Conductor 35283528
Sign −0.984−0.173i-0.984 - 0.173i
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  + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999i·6-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s + (0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + 17-s + (0.866 − 0.499i)18-s − 19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999i·6-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s + (0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + 17-s + (0.866 − 0.499i)18-s − 19-s + ⋯

Functional equation

Λ(s)=(3528s/2 ΓC(s) L(s)=((−0.984−0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3528s/2 ΓC(s) L(s)=((−0.984−0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35283528    =    23⋅32⋅722^{3} \cdot 3^{2} \cdot 7^{2}
Sign: −0.984−0.173i-0.984 - 0.173i
Analytic conductor: 1.760701.76070
Root analytic conductor: 1.326911.32691
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3528(2059,⋅)\chi_{3528} (2059, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3528, ( :0), −0.984−0.173i)(2,\ 3528,\ (\ :0),\ -0.984 - 0.173i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.20221884740.2022188474
L(12)L(\frac12) ≈\approx 0.20221884740.2022188474
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1 1
good5 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
11 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
17 1−T+T2 1 - T + T^{2}
19 1+T+T2 1 + T + T^{2}
23 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
29 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1−iT−T2 1 - iT - T^{2}
41 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+iT−T2 1 + iT - T^{2}
59 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+T+T2 1 + T + T^{2}
79 1+(1.73+i)T+(0.5+0.866i)T2 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2}
83 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
89 1+T+T2 1 + T + T^{2}
97 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)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

−8.369692084829507249966076536939, −7.56623737658236347202436486844, −7.09624325762129458868040765793, −6.47560255494082344491196512664, −5.53180956507558771160557584707, −4.29931682151293741919426009340, −3.39864557703299118438646358323, −2.55651692164476565764715583662, −1.49614180284212668885195459786, −0.18575091203530133148403206079, 1.27297274609380430027497668110, 2.77811340387787122638099571724, 3.96514661201432798844170104391, 4.70525641638961301599220273962, 5.37087692657741975151146833972, 6.15801425521371923824834134501, 7.15265739055284454859344791009, 7.63360244313688579326072212944, 8.490621689289729590995029150776, 9.174356066945332835880687283474

Graph of the ZZ-function along the critical line