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Properties

Label 2-567-189.101-c1-0-0
Degree 22
Conductor 567567
Sign 0.377+0.925i0.377 + 0.925i
Analytic cond. 4.527514.52751
Root an. cond. 2.127792.12779
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  + (−1.75 + 2.08i)2-s + (−0.940 − 5.33i)4-s + (−2.08 + 1.74i)5-s + (−0.122 + 2.64i)7-s + (8.05 + 4.65i)8-s − 7.40i·10-s + (−0.0911 + 0.108i)11-s + (−0.695 + 1.91i)13-s + (−5.29 − 4.88i)14-s + (−13.6 + 4.96i)16-s − 4.00·17-s + 3.04i·19-s + (11.2 + 9.47i)20-s + (−0.0670 − 0.380i)22-s + (0.449 − 1.23i)23-s + ⋯
L(s)  = 1  + (−1.23 + 1.47i)2-s + (−0.470 − 2.66i)4-s + (−0.931 + 0.781i)5-s + (−0.0463 + 0.998i)7-s + (2.84 + 1.64i)8-s − 2.34i·10-s + (−0.0274 + 0.0327i)11-s + (−0.192 + 0.530i)13-s + (−1.41 − 1.30i)14-s + (−3.40 + 1.24i)16-s − 0.970·17-s + 0.698i·19-s + (2.52 + 2.11i)20-s + (−0.0142 − 0.0810i)22-s + (0.0936 − 0.257i)23-s + ⋯

Functional equation

Λ(s)=(567s/2 ΓC(s) L(s)=((0.377+0.925i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(567s/2 ΓC(s+1/2) L(s)=((0.377+0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 567567    =    34⋅73^{4} \cdot 7
Sign: 0.377+0.925i0.377 + 0.925i
Analytic conductor: 4.527514.52751
Root analytic conductor: 2.127792.12779
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ567(143,⋅)\chi_{567} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 567, ( :1/2), 0.377+0.925i)(2,\ 567,\ (\ :1/2),\ 0.377 + 0.925i)

Particular Values

L(1)L(1) ≈\approx 0.0892191−0.0599668i0.0892191 - 0.0599668i
L(12)L(\frac12) ≈\approx 0.0892191−0.0599668i0.0892191 - 0.0599668i
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)
bad3 1 1
7 1+(0.122−2.64i)T 1 + (0.122 - 2.64i)T
good2 1+(1.75−2.08i)T+(−0.347−1.96i)T2 1 + (1.75 - 2.08i)T + (-0.347 - 1.96i)T^{2}
5 1+(2.08−1.74i)T+(0.868−4.92i)T2 1 + (2.08 - 1.74i)T + (0.868 - 4.92i)T^{2}
11 1+(0.0911−0.108i)T+(−1.91−10.8i)T2 1 + (0.0911 - 0.108i)T + (-1.91 - 10.8i)T^{2}
13 1+(0.695−1.91i)T+(−9.95−8.35i)T2 1 + (0.695 - 1.91i)T + (-9.95 - 8.35i)T^{2}
17 1+4.00T+17T2 1 + 4.00T + 17T^{2}
19 1−3.04iT−19T2 1 - 3.04iT - 19T^{2}
23 1+(−0.449+1.23i)T+(−17.6−14.7i)T2 1 + (-0.449 + 1.23i)T + (-17.6 - 14.7i)T^{2}
29 1+(2.12+5.84i)T+(−22.2+18.6i)T2 1 + (2.12 + 5.84i)T + (-22.2 + 18.6i)T^{2}
31 1+(4.18−0.738i)T+(29.1−10.6i)T2 1 + (4.18 - 0.738i)T + (29.1 - 10.6i)T^{2}
37 1+(−4.69+8.12i)T+(−18.5−32.0i)T2 1 + (-4.69 + 8.12i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.303+0.110i)T+(31.4+26.3i)T2 1 + (0.303 + 0.110i)T + (31.4 + 26.3i)T^{2}
43 1+(−0.643+3.65i)T+(−40.4−14.7i)T2 1 + (-0.643 + 3.65i)T + (-40.4 - 14.7i)T^{2}
47 1+(−1.15+6.57i)T+(−44.1−16.0i)T2 1 + (-1.15 + 6.57i)T + (-44.1 - 16.0i)T^{2}
53 1+(−6.59−3.81i)T+(26.5+45.8i)T2 1 + (-6.59 - 3.81i)T + (26.5 + 45.8i)T^{2}
59 1+(−6.06−2.20i)T+(45.1+37.9i)T2 1 + (-6.06 - 2.20i)T + (45.1 + 37.9i)T^{2}
61 1+(11.1+1.96i)T+(57.3+20.8i)T2 1 + (11.1 + 1.96i)T + (57.3 + 20.8i)T^{2}
67 1+(1.37−1.15i)T+(11.6−65.9i)T2 1 + (1.37 - 1.15i)T + (11.6 - 65.9i)T^{2}
71 1+(6.03−3.48i)T+(35.5−61.4i)T2 1 + (6.03 - 3.48i)T + (35.5 - 61.4i)T^{2}
73 1+(−3.67+2.12i)T+(36.5−63.2i)T2 1 + (-3.67 + 2.12i)T + (36.5 - 63.2i)T^{2}
79 1+(11.4+9.62i)T+(13.7+77.7i)T2 1 + (11.4 + 9.62i)T + (13.7 + 77.7i)T^{2}
83 1+(−0.966+0.351i)T+(63.5−53.3i)T2 1 + (-0.966 + 0.351i)T + (63.5 - 53.3i)T^{2}
89 1−5.17T+89T2 1 - 5.17T + 89T^{2}
97 1+(7.25+1.27i)T+(91.1+33.1i)T2 1 + (7.25 + 1.27i)T + (91.1 + 33.1i)T^{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

−11.15042232218033412631045024794, −10.36238602859588296536387830451, −9.301579212611109357175167314630, −8.754795298948647257692457347929, −7.80188228549080934272581032195, −7.17798249169070824952853598656, −6.31079091789113851960367470807, −5.49084631023550192122111227386, −4.16065636769253926974219420638, −2.17930707537428840748659508052, 0.10105334363817669444513759141, 1.27629067560258458213442240908, 2.90805127001682678691111185565, 3.97174010120991882834799659003, 4.71613312086504508600646169651, 7.00036920054260489366172917668, 7.74944065035013051763593577045, 8.483984535996241089125654123381, 9.238820316038258853504510053987, 10.10046047965778791477394946756

Graph of the ZZ-function along the critical line