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Properties

Label 2-320-5.4-c3-0-29
Degree 22
Conductor 320320
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 18.880618.8806
Root an. cond. 4.345184.34518
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  − 2i·3-s + (5 − 10i)5-s − 26i·7-s + 23·9-s + 28·11-s − 12i·13-s + (−20 − 10i)15-s + 64i·17-s − 60·19-s − 52·21-s − 58i·23-s + (−75 − 100i)25-s − 100i·27-s + 90·29-s − 128·31-s + ⋯
L(s)  = 1  − 0.384i·3-s + (0.447 − 0.894i)5-s − 1.40i·7-s + 0.851·9-s + 0.767·11-s − 0.256i·13-s + (−0.344 − 0.172i)15-s + 0.913i·17-s − 0.724·19-s − 0.540·21-s − 0.525i·23-s + (−0.599 − 0.800i)25-s − 0.712i·27-s + 0.576·29-s − 0.741·31-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+3/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{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: 320320    =    26⋅52^{6} \cdot 5
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 18.880618.8806
Root analytic conductor: 4.345184.34518
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ320(129,⋅)\chi_{320} (129, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 320, ( :3/2), −0.447+0.894i)(2,\ 320,\ (\ :3/2),\ -0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 2.0375912752.037591275
L(12)L(\frac12) ≈\approx 2.0375912752.037591275
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
5 1+(−5+10i)T 1 + (-5 + 10i)T
good3 1+2iT−27T2 1 + 2iT - 27T^{2}
7 1+26iT−343T2 1 + 26iT - 343T^{2}
11 1−28T+1.33e3T2 1 - 28T + 1.33e3T^{2}
13 1+12iT−2.19e3T2 1 + 12iT - 2.19e3T^{2}
17 1−64iT−4.91e3T2 1 - 64iT - 4.91e3T^{2}
19 1+60T+6.85e3T2 1 + 60T + 6.85e3T^{2}
23 1+58iT−1.21e4T2 1 + 58iT - 1.21e4T^{2}
29 1−90T+2.43e4T2 1 - 90T + 2.43e4T^{2}
31 1+128T+2.97e4T2 1 + 128T + 2.97e4T^{2}
37 1−236iT−5.06e4T2 1 - 236iT - 5.06e4T^{2}
41 1−242T+6.89e4T2 1 - 242T + 6.89e4T^{2}
43 1+362iT−7.95e4T2 1 + 362iT - 7.95e4T^{2}
47 1+226iT−1.03e5T2 1 + 226iT - 1.03e5T^{2}
53 1−108iT−1.48e5T2 1 - 108iT - 1.48e5T^{2}
59 1+20T+2.05e5T2 1 + 20T + 2.05e5T^{2}
61 1+542T+2.26e5T2 1 + 542T + 2.26e5T^{2}
67 1+434iT−3.00e5T2 1 + 434iT - 3.00e5T^{2}
71 1+1.12e3T+3.57e5T2 1 + 1.12e3T + 3.57e5T^{2}
73 1−632iT−3.89e5T2 1 - 632iT - 3.89e5T^{2}
79 1−720T+4.93e5T2 1 - 720T + 4.93e5T^{2}
83 1−478iT−5.71e5T2 1 - 478iT - 5.71e5T^{2}
89 1−490T+7.04e5T2 1 - 490T + 7.04e5T^{2}
97 1+1.45e3iT−9.12e5T2 1 + 1.45e3iT - 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

−10.65293564668183674781522240351, −10.11556071257465796326139075582, −8.995261770823402169029000634345, −8.010867569235655360101214641531, −7.00129800582575677014624815039, −6.13574437147212438980891751340, −4.61104182251339335596954819002, −3.89482963891029657179100245135, −1.75232921695408700796812412207, −0.77660257682822605928917914978, 1.84391091884897757701568395479, 3.02336676388724617027351559708, 4.39864792936923166987060598602, 5.68297296987992963923648045740, 6.55304683624049432613562631405, 7.56716214727069016926041410952, 9.119870743668240181451128152218, 9.436179251195188422269726643836, 10.57414412333405307026209586741, 11.44882135083684824148929030341

Graph of the ZZ-function along the critical line