↑

Properties

Label 2-960-5.4-c3-0-68
Degree 22
Conductor 960960
Sign −0.999−0.0160i-0.999 - 0.0160i
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 + (0.178 − 11.1i)5-s − 35.0i·7-s − 9·9-s − 25.6·11-s + 37.6i·13-s + (33.5 + 0.536i)15-s − 95.7i·17-s + 50.8·19-s + 105.·21-s − 110. i·23-s + (−124. − 4i)25-s − 27i·27-s − 54.5·29-s + 198.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.0160 − 0.999i)5-s − 1.89i·7-s − 0.333·9-s − 0.702·11-s + 0.803i·13-s + (0.577 + 0.00923i)15-s − 1.36i·17-s + 0.614·19-s + 1.09·21-s − 1.00i·23-s + (−0.999 − 0.0320i)25-s − 0.192i·27-s − 0.349·29-s + 1.14·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((−0.999−0.0160i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0160i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+3/2) L(s)=((−0.999−0.0160i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0160i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: −0.999−0.0160i-0.999 - 0.0160i
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), −0.999−0.0160i)(2,\ 960,\ (\ :3/2),\ -0.999 - 0.0160i)

Particular Values

L(2)L(2) ≈\approx 0.89709703070.8970970307
L(12)L(\frac12) ≈\approx 0.89709703070.8970970307
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+(−0.178+11.1i)T 1 + (-0.178 + 11.1i)T
good7 1+35.0iT−343T2 1 + 35.0iT - 343T^{2}
11 1+25.6T+1.33e3T2 1 + 25.6T + 1.33e3T^{2}
13 1−37.6iT−2.19e3T2 1 - 37.6iT - 2.19e3T^{2}
17 1+95.7iT−4.91e3T2 1 + 95.7iT - 4.91e3T^{2}
19 1−50.8T+6.85e3T2 1 - 50.8T + 6.85e3T^{2}
23 1+110.iT−1.21e4T2 1 + 110. iT - 1.21e4T^{2}
29 1+54.5T+2.43e4T2 1 + 54.5T + 2.43e4T^{2}
31 1−198.T+2.97e4T2 1 - 198.T + 2.97e4T^{2}
37 1+266.iT−5.06e4T2 1 + 266. iT - 5.06e4T^{2}
41 1−103.T+6.89e4T2 1 - 103.T + 6.89e4T^{2}
43 1−108iT−7.95e4T2 1 - 108iT - 7.95e4T^{2}
47 1−597.iT−1.03e5T2 1 - 597. iT - 1.03e5T^{2}
53 1+305.iT−1.48e5T2 1 + 305. iT - 1.48e5T^{2}
59 1+223.T+2.05e5T2 1 + 223.T + 2.05e5T^{2}
61 1+485.T+2.26e5T2 1 + 485.T + 2.26e5T^{2}
67 1−876.iT−3.00e5T2 1 - 876. iT - 3.00e5T^{2}
71 1−585.T+3.57e5T2 1 - 585.T + 3.57e5T^{2}
73 1−1.13e3iT−3.89e5T2 1 - 1.13e3iT - 3.89e5T^{2}
79 1+685.T+4.93e5T2 1 + 685.T + 4.93e5T^{2}
83 1−305.iT−5.71e5T2 1 - 305. iT - 5.71e5T^{2}
89 1+887.T+7.04e5T2 1 + 887.T + 7.04e5T^{2}
97 1+556.iT−9.12e5T2 1 + 556. iT - 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

−9.503107133563289856300618843674, −8.377817074078845434961107539890, −7.57506007086961423574320321824, −6.82173474593088831284943526072, −5.51080071241078138719220741387, −4.49852494971327762959357801024, −4.22197117415714484262611280425, −2.83315352689586680227219239077, −1.15710929027036725883454087288, −0.23820130380906611051906225258, 1.74797225225809062209640747076, 2.67808413607816863972215345814, 3.37492841695897209515294619006, 5.18663533659423481190396093977, 5.86423273286072830583365177404, 6.48523255203081336021150763756, 7.73784231777088848057390462844, 8.186515457468282068586006874569, 9.175080359184590366453340289597, 10.11266375889886116470986421652

Graph of the ZZ-function along the critical line