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Properties

Label 2-1920-120.59-c1-0-18
Degree 22
Conductor 19201920
Sign 0.912−0.408i0.912 - 0.408i
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 − 2.23·5-s − 5.23·7-s + (−2.23 − 2.00i)9-s + (−1.38 + 3.61i)15-s + (−3.23 + 8.47i)21-s + 5.70i·23-s + 5.00·25-s + (−4.61 + 2.38i)27-s + 6·29-s + 11.7·35-s − 12i·41-s + 11.2i·43-s + (5.00 + 4.47i)45-s − 13.7i·47-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s − 0.999·5-s − 1.97·7-s + (−0.745 − 0.666i)9-s + (−0.356 + 0.934i)15-s + (−0.706 + 1.84i)21-s + 1.19i·23-s + 1.00·25-s + (−0.888 + 0.458i)27-s + 1.11·29-s + 1.97·35-s − 1.87i·41-s + 1.71i·43-s + (0.745 + 0.666i)45-s − 1.99i·47-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((0.912−0.408i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s+1/2) L(s)=((0.912−0.408i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: 0.912−0.408i0.912 - 0.408i
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), 0.912−0.408i)(2,\ 1920,\ (\ :1/2),\ 0.912 - 0.408i)

Particular Values

L(1)L(1) ≈\approx 0.69208510870.6920851087
L(12)L(\frac12) ≈\approx 0.69208510870.6920851087
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+2.23T 1 + 2.23T
good7 1+5.23T+7T2 1 + 5.23T + 7T^{2}
11 1−11T2 1 - 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−5.70iT−23T2 1 - 5.70iT - 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+12iT−41T2 1 + 12iT - 41T^{2}
43 1−11.2iT−43T2 1 - 11.2iT - 43T^{2}
47 1+13.7iT−47T2 1 + 13.7iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1−59T2 1 - 59T^{2}
61 1−8iT−61T2 1 - 8iT - 61T^{2}
67 1−8.18iT−67T2 1 - 8.18iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1−4.29T+83T2 1 - 4.29T + 83T^{2}
89 1−17.8iT−89T2 1 - 17.8iT - 89T^{2}
97 1−97T2 1 - 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

−9.115365591252721180381405811777, −8.472325207204626714324109068252, −7.52714737963841010679996005428, −6.95704430847424740246822373561, −6.36251399546116178535706637469, −5.43591226437249346217464617614, −3.91808875249130013207707724221, −3.34431769883903162803898894090, −2.52506338656078348156080828548, −0.830386226316714201500292802031, 0.33406559510895863014558709123, 2.74350528394313331854383477052, 3.23747753694142801973158741297, 4.09567646597640512895996247776, 4.82499018857933670360990128283, 6.09605387708813105993272224185, 6.69026544663871064171687413278, 7.69608557192722666010814993555, 8.540204661585326246720798867002, 9.174790468012657029806259320400

Graph of the ZZ-function along the critical line