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Properties

Label 2-960-24.11-c3-0-13
Degree 22
Conductor 960960
Sign −0.999−0.0159i-0.999 - 0.0159i
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  + (3.61 + 3.73i)3-s + 5·5-s + 29.1i·7-s + (−0.862 + 26.9i)9-s − 37.6i·11-s − 58.2i·13-s + (18.0 + 18.6i)15-s + 73.3i·17-s − 95.1·19-s + (−108. + 105. i)21-s − 189.·23-s + 25·25-s + (−103. + 94.3i)27-s + 68.9·29-s + 264. i·31-s + ⋯
L(s)  = 1  + (0.695 + 0.718i)3-s + 0.447·5-s + 1.57i·7-s + (−0.0319 + 0.999i)9-s − 1.03i·11-s − 1.24i·13-s + (0.311 + 0.321i)15-s + 1.04i·17-s − 1.14·19-s + (−1.13 + 1.09i)21-s − 1.72·23-s + 0.200·25-s + (−0.740 + 0.672i)27-s + 0.441·29-s + 1.53i·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((−0.999−0.0159i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0159i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+3/2) L(s)=((−0.999−0.0159i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0159i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: −0.999−0.0159i-0.999 - 0.0159i
Analytic conductor: 56.641856.6418
Root analytic conductor: 7.526077.52607
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ960(671,⋅)\chi_{960} (671, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :3/2), −0.999−0.0159i)(2,\ 960,\ (\ :3/2),\ -0.999 - 0.0159i)

Particular Values

L(2)L(2) ≈\approx 1.5068518221.506851822
L(12)L(\frac12) ≈\approx 1.5068518221.506851822
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+(−3.61−3.73i)T 1 + (-3.61 - 3.73i)T
5 1−5T 1 - 5T
good7 1−29.1iT−343T2 1 - 29.1iT - 343T^{2}
11 1+37.6iT−1.33e3T2 1 + 37.6iT - 1.33e3T^{2}
13 1+58.2iT−2.19e3T2 1 + 58.2iT - 2.19e3T^{2}
17 1−73.3iT−4.91e3T2 1 - 73.3iT - 4.91e3T^{2}
19 1+95.1T+6.85e3T2 1 + 95.1T + 6.85e3T^{2}
23 1+189.T+1.21e4T2 1 + 189.T + 1.21e4T^{2}
29 1−68.9T+2.43e4T2 1 - 68.9T + 2.43e4T^{2}
31 1−264.iT−2.97e4T2 1 - 264. iT - 2.97e4T^{2}
37 1−338.iT−5.06e4T2 1 - 338. iT - 5.06e4T^{2}
41 1+145.iT−6.89e4T2 1 + 145. iT - 6.89e4T^{2}
43 1+51.9T+7.95e4T2 1 + 51.9T + 7.95e4T^{2}
47 1+173.T+1.03e5T2 1 + 173.T + 1.03e5T^{2}
53 1−212.T+1.48e5T2 1 - 212.T + 1.48e5T^{2}
59 1+699.iT−2.05e5T2 1 + 699. iT - 2.05e5T^{2}
61 1−292.iT−2.26e5T2 1 - 292. iT - 2.26e5T^{2}
67 1+919.T+3.00e5T2 1 + 919.T + 3.00e5T^{2}
71 1−563.T+3.57e5T2 1 - 563.T + 3.57e5T^{2}
73 1+11.2T+3.89e5T2 1 + 11.2T + 3.89e5T^{2}
79 1−680.iT−4.93e5T2 1 - 680. iT - 4.93e5T^{2}
83 1+696.iT−5.71e5T2 1 + 696. iT - 5.71e5T^{2}
89 1+1.64e3iT−7.04e5T2 1 + 1.64e3iT - 7.04e5T^{2}
97 1+437.T+9.12e5T2 1 + 437.T + 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

−10.21125488860315244937157217284, −9.041038573800162832041148683429, −8.391093636513222076446908087745, −8.142160308822461397553845980671, −6.31667740219769161595532100801, −5.74702051948455585336677307242, −4.89946504811180562258810703530, −3.57682885958326854348602605245, −2.77185807983605420315793285482, −1.82524176272505236250445245900, 0.30705335429984263348087330347, 1.65128619543445664016800485411, 2.40816526545133898897937674873, 3.99872456571999393800814452250, 4.40056228248010923041117845510, 6.08949462710938116548485799934, 6.90994222789457060507435739168, 7.39906595958231244694620134687, 8.247805692812581094896998729109, 9.419868032531057935245703291298

Graph of the ZZ-function along the critical line