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Properties

Label 2-1920-120.59-c1-0-64
Degree 22
Conductor 19201920
Sign 0.632+0.774i0.632 + 0.774i
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  + 1.73·3-s + (−1.73 + 1.41i)5-s − 2·7-s + 2.99·9-s − 5.65i·11-s + (−2.99 + 2.44i)15-s − 3.46·21-s + (0.999 − 4.89i)25-s + 5.19·27-s + 10.3·29-s − 4.89i·31-s − 9.79i·33-s + (3.46 − 2.82i)35-s + (−5.19 + 4.24i)45-s − 3·49-s + ⋯
L(s)  = 1  + 1.00·3-s + (−0.774 + 0.632i)5-s − 0.755·7-s + 0.999·9-s − 1.70i·11-s + (−0.774 + 0.632i)15-s − 0.755·21-s + (0.199 − 0.979i)25-s + 1.00·27-s + 1.92·29-s − 0.879i·31-s − 1.70i·33-s + (0.585 − 0.478i)35-s + (−0.774 + 0.632i)45-s − 0.428·49-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((0.632+0.774i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s+1/2) L(s)=((0.632+0.774i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: 0.632+0.774i0.632 + 0.774i
Analytic conductor: 15.331215.3312
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1920(959,⋅)\chi_{1920} (959, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :1/2), 0.632+0.774i)(2,\ 1920,\ (\ :1/2),\ 0.632 + 0.774i)

Particular Values

L(1)L(1) ≈\approx 1.8436887131.843688713
L(12)L(\frac12) ≈\approx 1.8436887131.843688713
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 1−1.73T 1 - 1.73T
5 1+(1.73−1.41i)T 1 + (1.73 - 1.41i)T
good7 1+2T+7T2 1 + 2T + 7T^{2}
11 1+5.65iT−11T2 1 + 5.65iT - 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1−10.3T+29T2 1 - 10.3T + 29T^{2}
31 1+4.89iT−31T2 1 + 4.89iT - 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1+14.1iT−53T2 1 + 14.1iT - 53T^{2}
59 1+11.3iT−59T2 1 + 11.3iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−9.79iT−73T2 1 - 9.79iT - 73T^{2}
79 1+14.6iT−79T2 1 + 14.6iT - 79T^{2}
83 1−17.3T+83T2 1 - 17.3T + 83T^{2}
89 1−89T2 1 - 89T^{2}
97 1−19.5iT−97T2 1 - 19.5iT - 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

−8.920549235071937807253042297190, −8.273770961203366508379190840850, −7.75742768395748844069041936062, −6.66614943306303814469330464034, −6.26413560208654578874956689615, −4.85305643680807661590647879360, −3.68293288472333325949891105179, −3.30229561026618056145331876431, −2.44700592499417615241129398976, −0.65038828366471350638881262615, 1.26320998212036097764306542801, 2.51932474252877980214338675371, 3.43903398399164501404732018528, 4.40793185243720684648719215900, 4.86342543598987039630628157073, 6.38840250001930732454950530171, 7.20128926716196058554084091440, 7.70985864118406660735799913545, 8.640769820100521849704441824558, 9.166978431314989588625088730227

Graph of the ZZ-function along the critical line