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Properties

Label 2-960-40.29-c3-0-32
Degree 22
Conductor 960960
Sign 0.988−0.153i0.988 - 0.153i
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  − 3·3-s + (−6.59 + 9.02i)5-s + 28.2i·7-s + 9·9-s − 23.3i·11-s + 7.81·13-s + (19.7 − 27.0i)15-s − 54.9i·17-s − 137. i·19-s − 84.8i·21-s + 58.1i·23-s + (−37.9 − 119. i)25-s − 27·27-s − 108. i·29-s − 117.·31-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.590 + 0.807i)5-s + 1.52i·7-s + 0.333·9-s − 0.641i·11-s + 0.166·13-s + (0.340 − 0.466i)15-s − 0.783i·17-s − 1.65i·19-s − 0.881i·21-s + 0.527i·23-s + (−0.303 − 0.952i)25-s − 0.192·27-s − 0.691i·29-s − 0.680·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.988−0.153i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+3/2) L(s)=((0.988−0.153i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.988−0.153i0.988 - 0.153i
Analytic conductor: 56.641856.6418
Root analytic conductor: 7.526077.52607
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ960(289,⋅)\chi_{960} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :3/2), 0.988−0.153i)(2,\ 960,\ (\ :3/2),\ 0.988 - 0.153i)

Particular Values

L(2)L(2) ≈\approx 1.2047054611.204705461
L(12)L(\frac12) ≈\approx 1.2047054611.204705461
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+3T 1 + 3T
5 1+(6.59−9.02i)T 1 + (6.59 - 9.02i)T
good7 1−28.2iT−343T2 1 - 28.2iT - 343T^{2}
11 1+23.3iT−1.33e3T2 1 + 23.3iT - 1.33e3T^{2}
13 1−7.81T+2.19e3T2 1 - 7.81T + 2.19e3T^{2}
17 1+54.9iT−4.91e3T2 1 + 54.9iT - 4.91e3T^{2}
19 1+137.iT−6.85e3T2 1 + 137. iT - 6.85e3T^{2}
23 1−58.1iT−1.21e4T2 1 - 58.1iT - 1.21e4T^{2}
29 1+108.iT−2.43e4T2 1 + 108. iT - 2.43e4T^{2}
31 1+117.T+2.97e4T2 1 + 117.T + 2.97e4T^{2}
37 1−62.2T+5.06e4T2 1 - 62.2T + 5.06e4T^{2}
41 1−69.0T+6.89e4T2 1 - 69.0T + 6.89e4T^{2}
43 1−456.T+7.95e4T2 1 - 456.T + 7.95e4T^{2}
47 1−326.iT−1.03e5T2 1 - 326. iT - 1.03e5T^{2}
53 1−61.7T+1.48e5T2 1 - 61.7T + 1.48e5T^{2}
59 1−641.iT−2.05e5T2 1 - 641. iT - 2.05e5T^{2}
61 1+818.iT−2.26e5T2 1 + 818. iT - 2.26e5T^{2}
67 1−493.T+3.00e5T2 1 - 493.T + 3.00e5T^{2}
71 1+669.T+3.57e5T2 1 + 669.T + 3.57e5T^{2}
73 1+14.8iT−3.89e5T2 1 + 14.8iT - 3.89e5T^{2}
79 1−1.06e3T+4.93e5T2 1 - 1.06e3T + 4.93e5T^{2}
83 1+1.02e3T+5.71e5T2 1 + 1.02e3T + 5.71e5T^{2}
89 1−1.07e3T+7.04e5T2 1 - 1.07e3T + 7.04e5T^{2}
97 1+69.1iT−9.12e5T2 1 + 69.1iT - 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

−9.509382247496534834160637482846, −8.970660575696004040168738601056, −7.913981343708345733344514051576, −7.05965253424353100914879663812, −6.15207232749671642281571855552, −5.48062338467604103336826041378, −4.42713033866703087666520064871, −3.11003139919101919996935569397, −2.36766306019893542050308799776, −0.51970014426095038170418484659, 0.72537124547935571739210808468, 1.65577485672724815242347972606, 3.76831220741109479209845304427, 4.14510289072269305541871130640, 5.16062612240399524763081909221, 6.18989360684978095382292079312, 7.25082200962248805795338789787, 7.77513157891826871843683373024, 8.707414101963792870360073387441, 9.830927970358815925071629848617

Graph of the ZZ-function along the critical line