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Properties

Label 2-1920-24.11-c1-0-56
Degree 22
Conductor 19201920
Sign −0.998−0.0549i-0.998 - 0.0549i
Analytic cond. 15.331215.3312
Root an. cond. 3.915513.91551
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15 − 1.29i)3-s − 5-s − 4.31i·7-s + (−0.329 + 2.98i)9-s − 5.96i·11-s − 4.89i·13-s + (1.15 + 1.29i)15-s − 1.07i·17-s + 4.23·19-s + (−5.56 + 4.98i)21-s + 5.38·23-s + 25-s + (4.22 − 3.02i)27-s + 2.80·29-s − 0.927i·31-s + ⋯
L(s)  = 1  + (−0.667 − 0.744i)3-s − 0.447·5-s − 1.62i·7-s + (−0.109 + 0.993i)9-s − 1.79i·11-s − 1.35i·13-s + (0.298 + 0.333i)15-s − 0.260i·17-s + 0.971·19-s + (−1.21 + 1.08i)21-s + 1.12·23-s + 0.200·25-s + (0.813 − 0.581i)27-s + 0.520·29-s − 0.166i·31-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((−0.998−0.0549i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s+1/2) L(s)=((−0.998−0.0549i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: −0.998−0.0549i-0.998 - 0.0549i
Analytic conductor: 15.331215.3312
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1920(191,⋅)\chi_{1920} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :1/2), −0.998−0.0549i)(2,\ 1920,\ (\ :1/2),\ -0.998 - 0.0549i)

Particular Values

L(1)L(1) ≈\approx 1.1077096481.107709648
L(12)L(\frac12) ≈\approx 1.1077096481.107709648
L(32)L(\frac{3}{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+(1.15+1.29i)T 1 + (1.15 + 1.29i)T
5 1+T 1 + T
good7 1+4.31iT−7T2 1 + 4.31iT - 7T^{2}
11 1+5.96iT−11T2 1 + 5.96iT - 11T^{2}
13 1+4.89iT−13T2 1 + 4.89iT - 13T^{2}
17 1+1.07iT−17T2 1 + 1.07iT - 17T^{2}
19 1−4.23T+19T2 1 - 4.23T + 19T^{2}
23 1−5.38T+23T2 1 - 5.38T + 23T^{2}
29 1−2.80T+29T2 1 - 2.80T + 29T^{2}
31 1+0.927iT−31T2 1 + 0.927iT - 31T^{2}
37 1+8.35iT−37T2 1 + 8.35iT - 37T^{2}
41 1−6.50iT−41T2 1 - 6.50iT - 41T^{2}
43 1+11.4T+43T2 1 + 11.4T + 43T^{2}
47 1−1.92T+47T2 1 - 1.92T + 47T^{2}
53 1−1.46T+53T2 1 - 1.46T + 53T^{2}
59 1−5.34iT−59T2 1 - 5.34iT - 59T^{2}
61 1+5.42iT−61T2 1 + 5.42iT - 61T^{2}
67 1−3.47T+67T2 1 - 3.47T + 67T^{2}
71 1−6.47T+71T2 1 - 6.47T + 71T^{2}
73 1−14.4T+73T2 1 - 14.4T + 73T^{2}
79 1+1.91iT−79T2 1 + 1.91iT - 79T^{2}
83 1−3.20iT−83T2 1 - 3.20iT - 83T^{2}
89 1−0.538iT−89T2 1 - 0.538iT - 89T^{2}
97 1+7.78T+97T2 1 + 7.78T + 97T^{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

−8.438444563164458425203171451351, −7.920237560113347185591863434972, −7.25152617497526954317850363476, −6.55291367651525928956868394095, −5.59025784724038575562810060172, −4.91258928838860113185593137370, −3.64090648598229838471637178930, −2.97981069460164402230497758233, −1.00637504819323588837761882412, −0.57169716556450281076206799637, 1.68647726734249860658071370337, 2.84953620721357525904735571481, 3.99655578847760957284909093443, 4.95206822273654236458614560625, 5.24584090681045802844229475883, 6.56097965965374842478493857947, 6.93312929107826766886767079999, 8.206490958502265327343273037999, 9.093723899139857710107787048037, 9.491522724393663471160387128067

Graph of the ZZ-function along the critical line