Properties

Label 2-546-91.41-c1-0-12
Degree 22
Conductor 546546
Sign 0.8870.460i0.887 - 0.460i
Analytic cond. 4.359834.35983
Root an. cond. 2.088022.08802
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  + (0.965 + 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.499i)4-s + (2.50 + 2.50i)5-s + (0.965 − 0.258i)6-s + (−1.87 − 1.86i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.77 + 3.06i)10-s + (−0.883 + 3.29i)11-s + 12-s + (1.87 − 3.08i)13-s + (−1.32 − 2.28i)14-s + (3.41 + 0.916i)15-s + (0.500 + 0.866i)16-s + (−0.906 + 1.57i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 + 0.249i)4-s + (1.11 + 1.11i)5-s + (0.394 − 0.105i)6-s + (−0.709 − 0.705i)7-s + (0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.559 + 0.969i)10-s + (−0.266 + 0.993i)11-s + 0.288·12-s + (0.518 − 0.854i)13-s + (−0.355 − 0.611i)14-s + (0.882 + 0.236i)15-s + (0.125 + 0.216i)16-s + (−0.219 + 0.380i)17-s + ⋯

Functional equation

Λ(s)=(546s/2ΓC(s)L(s)=((0.8870.460i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(546s/2ΓC(s+1/2)L(s)=((0.8870.460i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    237132 \cdot 3 \cdot 7 \cdot 13
Sign: 0.8870.460i0.887 - 0.460i
Analytic conductor: 4.359834.35983
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ546(223,)\chi_{546} (223, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 546, ( :1/2), 0.8870.460i)(2,\ 546,\ (\ :1/2),\ 0.887 - 0.460i)

Particular Values

L(1)L(1) \approx 2.72322+0.664759i2.72322 + 0.664759i
L(12)L(\frac12) \approx 2.72322+0.664759i2.72322 + 0.664759i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.9650.258i)T 1 + (-0.965 - 0.258i)T
3 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
7 1+(1.87+1.86i)T 1 + (1.87 + 1.86i)T
13 1+(1.87+3.08i)T 1 + (-1.87 + 3.08i)T
good5 1+(2.502.50i)T+5iT2 1 + (-2.50 - 2.50i)T + 5iT^{2}
11 1+(0.8833.29i)T+(9.525.5i)T2 1 + (0.883 - 3.29i)T + (-9.52 - 5.5i)T^{2}
17 1+(0.9061.57i)T+(8.514.7i)T2 1 + (0.906 - 1.57i)T + (-8.5 - 14.7i)T^{2}
19 1+(6.44+1.72i)T+(16.49.5i)T2 1 + (-6.44 + 1.72i)T + (16.4 - 9.5i)T^{2}
23 1+(5.863.38i)T+(11.519.9i)T2 1 + (5.86 - 3.38i)T + (11.5 - 19.9i)T^{2}
29 1+(2.24+3.87i)T+(14.5+25.1i)T2 1 + (2.24 + 3.87i)T + (-14.5 + 25.1i)T^{2}
31 1+(6.69+6.69i)T+31iT2 1 + (6.69 + 6.69i)T + 31iT^{2}
37 1+(0.544+2.03i)T+(32.018.5i)T2 1 + (-0.544 + 2.03i)T + (-32.0 - 18.5i)T^{2}
41 1+(1.435.35i)T+(35.520.5i)T2 1 + (1.43 - 5.35i)T + (-35.5 - 20.5i)T^{2}
43 1+(2.23+1.28i)T+(21.5+37.2i)T2 1 + (2.23 + 1.28i)T + (21.5 + 37.2i)T^{2}
47 1+(2.09+2.09i)T47iT2 1 + (-2.09 + 2.09i)T - 47iT^{2}
53 112.2T+53T2 1 - 12.2T + 53T^{2}
59 1+(0.2020.753i)T+(51.0+29.5i)T2 1 + (-0.202 - 0.753i)T + (-51.0 + 29.5i)T^{2}
61 1+(3.49+2.01i)T+(30.5+52.8i)T2 1 + (3.49 + 2.01i)T + (30.5 + 52.8i)T^{2}
67 1+(11.6+3.13i)T+(58.0+33.5i)T2 1 + (11.6 + 3.13i)T + (58.0 + 33.5i)T^{2}
71 1+(1.20+4.48i)T+(61.4+35.5i)T2 1 + (1.20 + 4.48i)T + (-61.4 + 35.5i)T^{2}
73 1+(4.59+4.59i)T73iT2 1 + (-4.59 + 4.59i)T - 73iT^{2}
79 18.32T+79T2 1 - 8.32T + 79T^{2}
83 1+(10.8+10.8i)T+83iT2 1 + (10.8 + 10.8i)T + 83iT^{2}
89 1+(6.87+1.84i)T+(77.0+44.5i)T2 1 + (6.87 + 1.84i)T + (77.0 + 44.5i)T^{2}
97 1+(1.49+0.401i)T+(84.048.5i)T2 1 + (-1.49 + 0.401i)T + (84.0 - 48.5i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.71925853176903160000809746875, −9.997346290600912569211371597114, −9.418557374564260978183682141890, −7.70128592579457937486466467114, −7.25669084453866307288231424364, −6.24917261075135170315303712148, −5.56904774413443644412325455171, −3.92264709062449569988600305156, −3.02777960833541359769789408250, −1.96379759232745674623012129776, 1.59379829682392479623482979330, 2.83897002556809528450419081902, 3.96023697865023473115282282827, 5.35628654830092438590245660845, 5.69127632321221425224722948128, 6.85826877138938558780540679562, 8.423744143211347630125075785127, 9.074087375768042296945535548290, 9.698067008246533692061644106251, 10.66117070810307229771494944844

Graph of the ZZ-function along the critical line