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Properties

Label 2-960-5.4-c3-0-52
Degree 22
Conductor 960960
Sign 0.447+0.894i0.447 + 0.894i
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 + (10 − 5i)5-s − 10i·7-s − 9·9-s + 46·11-s + 34i·13-s + (−15 − 30i)15-s − 66i·17-s + 104·19-s − 30·21-s + 164i·23-s + (75 − 100i)25-s + 27i·27-s + 224·29-s − 72·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.894 − 0.447i)5-s − 0.539i·7-s − 0.333·9-s + 1.26·11-s + 0.725i·13-s + (−0.258 − 0.516i)15-s − 0.941i·17-s + 1.25·19-s − 0.311·21-s + 1.48i·23-s + (0.599 − 0.800i)25-s + 0.192i·27-s + 1.43·29-s − 0.417·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+3/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.447+0.894i0.447 + 0.894i
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), 0.447+0.894i)(2,\ 960,\ (\ :3/2),\ 0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 2.9232295342.923229534
L(12)L(\frac12) ≈\approx 2.9232295342.923229534
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+(−10+5i)T 1 + (-10 + 5i)T
good7 1+10iT−343T2 1 + 10iT - 343T^{2}
11 1−46T+1.33e3T2 1 - 46T + 1.33e3T^{2}
13 1−34iT−2.19e3T2 1 - 34iT - 2.19e3T^{2}
17 1+66iT−4.91e3T2 1 + 66iT - 4.91e3T^{2}
19 1−104T+6.85e3T2 1 - 104T + 6.85e3T^{2}
23 1−164iT−1.21e4T2 1 - 164iT - 1.21e4T^{2}
29 1−224T+2.43e4T2 1 - 224T + 2.43e4T^{2}
31 1+72T+2.97e4T2 1 + 72T + 2.97e4T^{2}
37 1+22iT−5.06e4T2 1 + 22iT - 5.06e4T^{2}
41 1−194T+6.89e4T2 1 - 194T + 6.89e4T^{2}
43 1+108iT−7.95e4T2 1 + 108iT - 7.95e4T^{2}
47 1−480iT−1.03e5T2 1 - 480iT - 1.03e5T^{2}
53 1+286iT−1.48e5T2 1 + 286iT - 1.48e5T^{2}
59 1−426T+2.05e5T2 1 - 426T + 2.05e5T^{2}
61 1+698T+2.26e5T2 1 + 698T + 2.26e5T^{2}
67 1−328iT−3.00e5T2 1 - 328iT - 3.00e5T^{2}
71 1−188T+3.57e5T2 1 - 188T + 3.57e5T^{2}
73 1+740iT−3.89e5T2 1 + 740iT - 3.89e5T^{2}
79 1+1.16e3T+4.93e5T2 1 + 1.16e3T + 4.93e5T^{2}
83 1+412iT−5.71e5T2 1 + 412iT - 5.71e5T^{2}
89 1+1.20e3T+7.04e5T2 1 + 1.20e3T + 7.04e5T^{2}
97 1−1.38e3iT−9.12e5T2 1 - 1.38e3iT - 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

−9.359856705317194039034172823374, −8.929821366979139158928071203640, −7.62817311571429252907055490275, −6.96046497984652316455803950632, −6.13924826458358244956256096774, −5.22474665674629472980698275071, −4.19129528310316115391884909280, −2.94929778838952601883422259703, −1.59800947319570399613448394829, −0.923918485388704903815822943164, 1.10666592587872645003552225731, 2.44913665195479619226571796935, 3.37420412480245023801702835326, 4.50991321428618044809814007283, 5.60250513752206878391347172866, 6.19671970509778178056173638606, 7.09181458801760521619867815219, 8.420437932161266074418861757186, 8.998086635960919481911835006963, 9.902919696566186985700016688188

Graph of the ZZ-function along the critical line