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Properties

Label 2-960-5.4-c3-0-11
Degree 22
Conductor 960960
Sign −0.116−0.993i-0.116 - 0.993i
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 + 1.29i)5-s − 16.2i·7-s − 9·9-s − 40.2·11-s − 19.7i·13-s + (−3.89 − 33.3i)15-s − 83.0i·17-s + 48.8·19-s + 48.6·21-s − 1.61i·23-s + (121. − 28.8i)25-s − 27i·27-s − 24.5·29-s + 12.4·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.993 + 0.116i)5-s − 0.875i·7-s − 0.333·9-s − 1.10·11-s − 0.422i·13-s + (−0.0670 − 0.573i)15-s − 1.18i·17-s + 0.589·19-s + 0.505·21-s − 0.0146i·23-s + (0.973 − 0.230i)25-s − 0.192i·27-s − 0.157·29-s + 0.0719·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((−0.116−0.993i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+3/2) L(s)=((−0.116−0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: −0.116−0.993i-0.116 - 0.993i
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), −0.116−0.993i)(2,\ 960,\ (\ :3/2),\ -0.116 - 0.993i)

Particular Values

L(2)L(2) ≈\approx 0.73988361580.7398836158
L(12)L(\frac12) ≈\approx 0.73988361580.7398836158
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+(11.1−1.29i)T 1 + (11.1 - 1.29i)T
good7 1+16.2iT−343T2 1 + 16.2iT - 343T^{2}
11 1+40.2T+1.33e3T2 1 + 40.2T + 1.33e3T^{2}
13 1+19.7iT−2.19e3T2 1 + 19.7iT - 2.19e3T^{2}
17 1+83.0iT−4.91e3T2 1 + 83.0iT - 4.91e3T^{2}
19 1−48.8T+6.85e3T2 1 - 48.8T + 6.85e3T^{2}
23 1+1.61iT−1.21e4T2 1 + 1.61iT - 1.21e4T^{2}
29 1+24.5T+2.43e4T2 1 + 24.5T + 2.43e4T^{2}
31 1−12.4T+2.97e4T2 1 - 12.4T + 2.97e4T^{2}
37 1−325.iT−5.06e4T2 1 - 325. iT - 5.06e4T^{2}
41 1+242.T+6.89e4T2 1 + 242.T + 6.89e4T^{2}
43 1−367.iT−7.95e4T2 1 - 367. iT - 7.95e4T^{2}
47 1+204.iT−1.03e5T2 1 + 204. iT - 1.03e5T^{2}
53 1−61.5iT−1.48e5T2 1 - 61.5iT - 1.48e5T^{2}
59 1−112.T+2.05e5T2 1 - 112.T + 2.05e5T^{2}
61 1+477.T+2.26e5T2 1 + 477.T + 2.26e5T^{2}
67 1+558.iT−3.00e5T2 1 + 558. iT - 3.00e5T^{2}
71 1+558.T+3.57e5T2 1 + 558.T + 3.57e5T^{2}
73 1−1.01e3iT−3.89e5T2 1 - 1.01e3iT - 3.89e5T^{2}
79 1−1.15e3T+4.93e5T2 1 - 1.15e3T + 4.93e5T^{2}
83 1+1.15e3iT−5.71e5T2 1 + 1.15e3iT - 5.71e5T^{2}
89 1+96.9T+7.04e5T2 1 + 96.9T + 7.04e5T^{2}
97 1−1.15e3iT−9.12e5T2 1 - 1.15e3iT - 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

−10.08845511056450179096439683863, −9.063668819743665929497923799160, −7.975465943201648850639631464684, −7.57654437412044950683577284029, −6.60461857970188157464548023136, −5.19805131087996443757181601055, −4.63897666143510660975661492301, −3.52424259675361632723343513281, −2.80682396021086534671052688076, −0.810534400152199493417260728516, 0.25631775832331570205085822254, 1.80484286687918247020763659406, 2.89842614835947036478143944502, 3.97110719126549774277080327935, 5.16172532638493311697288665766, 5.91794218707413854285266948417, 7.04579010513617666373750870874, 7.78516268670009788251395143883, 8.468444708679054963411658983035, 9.163300974027634675921210679798

Graph of the ZZ-function along the critical line