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Properties

Label 2-960-5.4-c3-0-18
Degree 22
Conductor 960960
Sign 0.894−0.447i0.894 - 0.447i
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 + (−5 − 10i)5-s + 4i·7-s − 9·9-s + 28·11-s − 16i·13-s + (−30 + 15i)15-s + 108i·17-s + 32·19-s + 12·21-s + 28i·23-s + (−75 + 100i)25-s + 27i·27-s − 238·29-s − 180·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.447 − 0.894i)5-s + 0.215i·7-s − 0.333·9-s + 0.767·11-s − 0.341i·13-s + (−0.516 + 0.258i)15-s + 1.54i·17-s + 0.386·19-s + 0.124·21-s + 0.253i·23-s + (−0.599 + 0.800i)25-s + 0.192i·27-s − 1.52·29-s − 1.04·31-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.894−0.447i0.894 - 0.447i
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), 0.894−0.447i)(2,\ 960,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.4105411201.410541120
L(12)L(\frac12) ≈\approx 1.4105411201.410541120
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+(5+10i)T 1 + (5 + 10i)T
good7 1−4iT−343T2 1 - 4iT - 343T^{2}
11 1−28T+1.33e3T2 1 - 28T + 1.33e3T^{2}
13 1+16iT−2.19e3T2 1 + 16iT - 2.19e3T^{2}
17 1−108iT−4.91e3T2 1 - 108iT - 4.91e3T^{2}
19 1−32T+6.85e3T2 1 - 32T + 6.85e3T^{2}
23 1−28iT−1.21e4T2 1 - 28iT - 1.21e4T^{2}
29 1+238T+2.43e4T2 1 + 238T + 2.43e4T^{2}
31 1+180T+2.97e4T2 1 + 180T + 2.97e4T^{2}
37 1−40iT−5.06e4T2 1 - 40iT - 5.06e4T^{2}
41 1−422T+6.89e4T2 1 - 422T + 6.89e4T^{2}
43 1−276iT−7.95e4T2 1 - 276iT - 7.95e4T^{2}
47 1−60iT−1.03e5T2 1 - 60iT - 1.03e5T^{2}
53 1−220iT−1.48e5T2 1 - 220iT - 1.48e5T^{2}
59 1+804T+2.05e5T2 1 + 804T + 2.05e5T^{2}
61 1−358T+2.26e5T2 1 - 358T + 2.26e5T^{2}
67 1−884iT−3.00e5T2 1 - 884iT - 3.00e5T^{2}
71 1+64T+3.57e5T2 1 + 64T + 3.57e5T^{2}
73 1−152iT−3.89e5T2 1 - 152iT - 3.89e5T^{2}
79 1−932T+4.93e5T2 1 - 932T + 4.93e5T^{2}
83 1+1.29e3iT−5.71e5T2 1 + 1.29e3iT - 5.71e5T^{2}
89 1−1.14e3T+7.04e5T2 1 - 1.14e3T + 7.04e5T^{2}
97 1−824iT−9.12e5T2 1 - 824iT - 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

−9.375669761276239025467676266918, −8.921617590435870099819280121792, −7.940231604187553626058958271784, −7.40581940253367705944549159150, −6.13658793907514267403701130081, −5.54396947160107077586116768728, −4.29680459460580734505408145264, −3.47767673272146598574600325041, −1.90944381370871878544777922199, −0.993314128233941416643496697739, 0.42363763931967096986831234987, 2.20491403632821746159877990550, 3.38026130916138417331494975619, 4.05114616346553013144725261851, 5.13972110641544922493447694840, 6.20349161443271520272515510734, 7.17358872072716866533678592072, 7.66939592366608689363062534726, 9.088790357346323675094785603966, 9.437970580805497472742977447913

Graph of the ZZ-function along the critical line