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Properties

Label 2-960-5.4-c3-0-71
Degree 22
Conductor 960960
Sign 0.0160−0.999i0.0160 - 0.999i
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + (−11.1 − 0.178i)5-s − 33.0i·7-s − 9·9-s − 48.3·11-s − 60.3i·13-s + (−0.536 + 33.5i)15-s − 17.7i·17-s − 130.·19-s − 99.2·21-s − 70.8i·23-s + (124. + 4i)25-s + 27i·27-s + 104.·29-s − 210.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.999 − 0.0160i)5-s − 1.78i·7-s − 0.333·9-s − 1.32·11-s − 1.28i·13-s + (−0.00923 + 0.577i)15-s − 0.253i·17-s − 1.58·19-s − 1.03·21-s − 0.642i·23-s + (0.999 + 0.0320i)25-s + 0.192i·27-s + 0.669·29-s − 1.21·31-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.0160−0.999i0.0160 - 0.999i
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), 0.0160−0.999i)(2,\ 960,\ (\ :3/2),\ 0.0160 - 0.999i)

Particular Values

L(2)L(2) ≈\approx 0.48202534110.4820253411
L(12)L(\frac12) ≈\approx 0.48202534110.4820253411
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+(11.1+0.178i)T 1 + (11.1 + 0.178i)T
good7 1+33.0iT−343T2 1 + 33.0iT - 343T^{2}
11 1+48.3T+1.33e3T2 1 + 48.3T + 1.33e3T^{2}
13 1+60.3iT−2.19e3T2 1 + 60.3iT - 2.19e3T^{2}
17 1+17.7iT−4.91e3T2 1 + 17.7iT - 4.91e3T^{2}
19 1+130.T+6.85e3T2 1 + 130.T + 6.85e3T^{2}
23 1+70.8iT−1.21e4T2 1 + 70.8iT - 1.21e4T^{2}
29 1−104.T+2.43e4T2 1 - 104.T + 2.43e4T^{2}
31 1+210.T+2.97e4T2 1 + 210.T + 2.97e4T^{2}
37 1+300.iT−5.06e4T2 1 + 300. iT - 5.06e4T^{2}
41 1−240.T+6.89e4T2 1 - 240.T + 6.89e4T^{2}
43 1+108iT−7.95e4T2 1 + 108iT - 7.95e4T^{2}
47 1+278.iT−1.03e5T2 1 + 278. iT - 1.03e5T^{2}
53 1−328.iT−1.48e5T2 1 - 328. iT - 1.48e5T^{2}
59 1−889.T+2.05e5T2 1 - 889.T + 2.05e5T^{2}
61 1−241.T+2.26e5T2 1 - 241.T + 2.26e5T^{2}
67 1+103.iT−3.00e5T2 1 + 103. iT - 3.00e5T^{2}
71 1+277.T+3.57e5T2 1 + 277.T + 3.57e5T^{2}
73 1+274.iT−3.89e5T2 1 + 274. iT - 3.89e5T^{2}
79 1+366.T+4.93e5T2 1 + 366.T + 4.93e5T^{2}
83 1−57.7iT−5.71e5T2 1 - 57.7iT - 5.71e5T^{2}
89 1−203.T+7.04e5T2 1 - 203.T + 7.04e5T^{2}
97 1−1.28e3iT−9.12e5T2 1 - 1.28e3iT - 9.12e5T^{2}
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   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

−8.684864439190878090450450222531, −7.937367687718877388300769067852, −7.45324901334651588087071511638, −6.76758807636533982879548005677, −5.48109088318667325212095964016, −4.42111742583327510673412378804, −3.60854358608341440167908466835, −2.46070596901356179618209060246, −0.69979004844029313231538163536, −0.18227004539074253846466594989, 2.08137921179789906170807221378, 2.95578159069212447572066193867, 4.15979795246281205144207086746, 4.98020717173412635681943499803, 5.85190979733750193427746935887, 6.84508754555273891502008136429, 8.089543048920087776526940778248, 8.572540346350207560321510522074, 9.282950702873196273193481387472, 10.27977306818854730957052407056

Graph of the ZZ-function along the critical line