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Properties

Label 2-1920-12.11-c1-0-19
Degree 22
Conductor 19201920
Sign 0.356−0.934i0.356 - 0.934i
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  + (−0.618 + 1.61i)3-s + i·5-s − 2i·7-s + (−2.23 − 2.00i)9-s + 2.47·11-s + 1.23·13-s + (−1.61 − 0.618i)15-s − 0.763i·17-s + 5.23i·19-s + (3.23 + 1.23i)21-s + 0.472·23-s − 25-s + (4.61 − 2.38i)27-s − 8.47i·29-s + 4.76i·31-s + ⋯
L(s)  = 1  + (−0.356 + 0.934i)3-s + 0.447i·5-s − 0.755i·7-s + (−0.745 − 0.666i)9-s + 0.745·11-s + 0.342·13-s + (−0.417 − 0.159i)15-s − 0.185i·17-s + 1.20i·19-s + (0.706 + 0.269i)21-s + 0.0984·23-s − 0.200·25-s + (0.888 − 0.458i)27-s − 1.57i·29-s + 0.855i·31-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((0.356−0.934i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s+1/2) L(s)=((0.356−0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: 0.356−0.934i0.356 - 0.934i
Analytic conductor: 15.331215.3312
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1920(1151,⋅)\chi_{1920} (1151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :1/2), 0.356−0.934i)(2,\ 1920,\ (\ :1/2),\ 0.356 - 0.934i)

Particular Values

L(1)L(1) ≈\approx 1.5246584991.524658499
L(12)L(\frac12) ≈\approx 1.5246584991.524658499
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+(0.618−1.61i)T 1 + (0.618 - 1.61i)T
5 1−iT 1 - iT
good7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1−2.47T+11T2 1 - 2.47T + 11T^{2}
13 1−1.23T+13T2 1 - 1.23T + 13T^{2}
17 1+0.763iT−17T2 1 + 0.763iT - 17T^{2}
19 1−5.23iT−19T2 1 - 5.23iT - 19T^{2}
23 1−0.472T+23T2 1 - 0.472T + 23T^{2}
29 1+8.47iT−29T2 1 + 8.47iT - 29T^{2}
31 1−4.76iT−31T2 1 - 4.76iT - 31T^{2}
37 1−7.70T+37T2 1 - 7.70T + 37T^{2}
41 1+1.52iT−41T2 1 + 1.52iT - 41T^{2}
43 1−9.70iT−43T2 1 - 9.70iT - 43T^{2}
47 1−4.47T+47T2 1 - 4.47T + 47T^{2}
53 1+4.47iT−53T2 1 + 4.47iT - 53T^{2}
59 1−6.47T+59T2 1 - 6.47T + 59T^{2}
61 1−12.4T+61T2 1 - 12.4T + 61T^{2}
67 1−11.2iT−67T2 1 - 11.2iT - 67T^{2}
71 1−4T+71T2 1 - 4T + 71T^{2}
73 1+0.472T+73T2 1 + 0.472T + 73T^{2}
79 1−8.18iT−79T2 1 - 8.18iT - 79T^{2}
83 1+11.7T+83T2 1 + 11.7T + 83T^{2}
89 1−1.52iT−89T2 1 - 1.52iT - 89T^{2}
97 1+12.4T+97T2 1 + 12.4T + 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

−9.687963597511723098973860301978, −8.623038536613229651034898456164, −7.85901279630310534247801194533, −6.82398750904941092109139357871, −6.16131530400902620773641473818, −5.35177240514990307255297511844, −4.08566541195703151691175913440, −3.90630619031656235645132869779, −2.66898222692323206581130604943, −1.01223335450960152705044685934, 0.75487618433219003998161834055, 1.89337015430623446756137943058, 2.87778221110115695290435046549, 4.18389345280658754818252238672, 5.24606143271271431790127939873, 5.85131608777129323898687340792, 6.72195971696560272052635887975, 7.34359476367446198920440958652, 8.432095716789926301089049614303, 8.839107277052600642613345239342

Graph of the ZZ-function along the critical line