↑

Properties

Label 2-960-5.4-c3-0-40
Degree 22
Conductor 960960
Sign 0.983+0.178i0.983 + 0.178i
Analytic cond. 56.641856.6418
Root an. cond. 7.526077.52607
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + (−2 + 11i)5-s − 2i·7-s − 9·9-s + 70·11-s − 54i·13-s + (33 + 6i)15-s + 22i·17-s − 24·19-s − 6·21-s + 100i·23-s + (−117 − 44i)25-s + 27i·27-s + 216·29-s − 208·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.178 + 0.983i)5-s − 0.107i·7-s − 0.333·9-s + 1.91·11-s − 1.15i·13-s + (0.568 + 0.103i)15-s + 0.313i·17-s − 0.289·19-s − 0.0623·21-s + 0.906i·23-s + (−0.936 − 0.351i)25-s + 0.192i·27-s + 1.38·29-s − 1.20·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.983+0.178i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+3/2) L(s)=((0.983+0.178i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.983+0.178i0.983 + 0.178i
Analytic conductor: 56.641856.6418
Root analytic conductor: 7.526077.52607
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ960(769,⋅)\chi_{960} (769, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :3/2), 0.983+0.178i)(2,\ 960,\ (\ :3/2),\ 0.983 + 0.178i)

Particular Values

L(2)L(2) ≈\approx 2.1785995322.178599532
L(12)L(\frac12) ≈\approx 2.1785995322.178599532
L(52)L(\frac{5}{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
3 1+3iT 1 + 3iT
5 1+(2−11i)T 1 + (2 - 11i)T
good7 1+2iT−343T2 1 + 2iT - 343T^{2}
11 1−70T+1.33e3T2 1 - 70T + 1.33e3T^{2}
13 1+54iT−2.19e3T2 1 + 54iT - 2.19e3T^{2}
17 1−22iT−4.91e3T2 1 - 22iT - 4.91e3T^{2}
19 1+24T+6.85e3T2 1 + 24T + 6.85e3T^{2}
23 1−100iT−1.21e4T2 1 - 100iT - 1.21e4T^{2}
29 1−216T+2.43e4T2 1 - 216T + 2.43e4T^{2}
31 1+208T+2.97e4T2 1 + 208T + 2.97e4T^{2}
37 1+254iT−5.06e4T2 1 + 254iT - 5.06e4T^{2}
41 1+206T+6.89e4T2 1 + 206T + 6.89e4T^{2}
43 1−292iT−7.95e4T2 1 - 292iT - 7.95e4T^{2}
47 1+320iT−1.03e5T2 1 + 320iT - 1.03e5T^{2}
53 1−402iT−1.48e5T2 1 - 402iT - 1.48e5T^{2}
59 1−370T+2.05e5T2 1 - 370T + 2.05e5T^{2}
61 1−550T+2.26e5T2 1 - 550T + 2.26e5T^{2}
67 1+728iT−3.00e5T2 1 + 728iT - 3.00e5T^{2}
71 1−540T+3.57e5T2 1 - 540T + 3.57e5T^{2}
73 1−604iT−3.89e5T2 1 - 604iT - 3.89e5T^{2}
79 1−792T+4.93e5T2 1 - 792T + 4.93e5T^{2}
83 1−404iT−5.71e5T2 1 - 404iT - 5.71e5T^{2}
89 1−938T+7.04e5T2 1 - 938T + 7.04e5T^{2}
97 1+56iT−9.12e5T2 1 + 56iT - 9.12e5T^{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

−9.640015039089004751355671340319, −8.717425008929317773583590780597, −7.81333637005420609414859209230, −6.99164220618287742172170881065, −6.39808282576933830047452847566, −5.51420709992944018050773008002, −3.98150145691963405328867845134, −3.29874965429420689782219854862, −2.02981178325438252640846382049, −0.808528621866258996298362676715, 0.828121072329971548702486123571, 1.99198318130975708246336157194, 3.65898375533114537101263828948, 4.31465635229975468909707521738, 5.07533428766961604031997568931, 6.30600560368817961417891502801, 6.95119104737054049554836208855, 8.382032193974636905220141939523, 8.916077389744112995404585936578, 9.443027152839833840281802098872

Graph of the ZZ-function along the critical line