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Properties

Label 2-666-111.14-c1-0-4
Degree 22
Conductor 666666
Sign 0.600−0.799i0.600 - 0.799i
Analytic cond. 5.318035.31803
Root an. cond. 2.306082.30608
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.205 − 0.765i)5-s + (−0.103 − 0.179i)7-s + (−0.707 − 0.707i)8-s + 0.792·10-s + 2.15·11-s + (3.34 + 0.896i)13-s + (0.146 − 0.146i)14-s + (0.500 − 0.866i)16-s + (−1.47 + 0.394i)17-s + (5.53 + 1.48i)19-s + (0.205 + 0.765i)20-s + (0.557 + 2.08i)22-s + (−0.186 − 0.186i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.0917 − 0.342i)5-s + (−0.0392 − 0.0679i)7-s + (−0.249 − 0.249i)8-s + 0.250·10-s + 0.649·11-s + (0.927 + 0.248i)13-s + (0.0392 − 0.0392i)14-s + (0.125 − 0.216i)16-s + (−0.357 + 0.0956i)17-s + (1.26 + 0.339i)19-s + (0.0458 + 0.171i)20-s + (0.118 + 0.443i)22-s + (−0.0388 − 0.0388i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((0.600−0.799i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((0.600−0.799i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 0.600−0.799i0.600 - 0.799i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(125,⋅)\chi_{666} (125, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), 0.600−0.799i)(2,\ 666,\ (\ :1/2),\ 0.600 - 0.799i)

Particular Values

L(1)L(1) ≈\approx 1.52901+0.763744i1.52901 + 0.763744i
L(12)L(\frac12) ≈\approx 1.52901+0.763744i1.52901 + 0.763744i
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+(−0.258−0.965i)T 1 + (-0.258 - 0.965i)T
3 1 1
37 1+(−5.85−1.66i)T 1 + (-5.85 - 1.66i)T
good5 1+(−0.205+0.765i)T+(−4.33−2.5i)T2 1 + (-0.205 + 0.765i)T + (-4.33 - 2.5i)T^{2}
7 1+(0.103+0.179i)T+(−3.5+6.06i)T2 1 + (0.103 + 0.179i)T + (-3.5 + 6.06i)T^{2}
11 1−2.15T+11T2 1 - 2.15T + 11T^{2}
13 1+(−3.34−0.896i)T+(11.2+6.5i)T2 1 + (-3.34 - 0.896i)T + (11.2 + 6.5i)T^{2}
17 1+(1.47−0.394i)T+(14.7−8.5i)T2 1 + (1.47 - 0.394i)T + (14.7 - 8.5i)T^{2}
19 1+(−5.53−1.48i)T+(16.4+9.5i)T2 1 + (-5.53 - 1.48i)T + (16.4 + 9.5i)T^{2}
23 1+(0.186+0.186i)T+23iT2 1 + (0.186 + 0.186i)T + 23iT^{2}
29 1+(0.667−0.667i)T−29iT2 1 + (0.667 - 0.667i)T - 29iT^{2}
31 1+(−2.39−2.39i)T+31iT2 1 + (-2.39 - 2.39i)T + 31iT^{2}
41 1+(2.63+4.55i)T+(−20.5+35.5i)T2 1 + (2.63 + 4.55i)T + (-20.5 + 35.5i)T^{2}
43 1+(1.86−1.86i)T−43iT2 1 + (1.86 - 1.86i)T - 43iT^{2}
47 1−3.73iT−47T2 1 - 3.73iT - 47T^{2}
53 1+(0.970+0.560i)T+(26.5+45.8i)T2 1 + (0.970 + 0.560i)T + (26.5 + 45.8i)T^{2}
59 1+(6.83−1.83i)T+(51.0−29.5i)T2 1 + (6.83 - 1.83i)T + (51.0 - 29.5i)T^{2}
61 1+(0.320−1.19i)T+(−52.8−30.5i)T2 1 + (0.320 - 1.19i)T + (-52.8 - 30.5i)T^{2}
67 1+(6.44−3.72i)T+(33.5−58.0i)T2 1 + (6.44 - 3.72i)T + (33.5 - 58.0i)T^{2}
71 1+(−3.47+2.00i)T+(35.5−61.4i)T2 1 + (-3.47 + 2.00i)T + (35.5 - 61.4i)T^{2}
73 1+9.83iT−73T2 1 + 9.83iT - 73T^{2}
79 1+(3.34+0.896i)T+(68.4+39.5i)T2 1 + (3.34 + 0.896i)T + (68.4 + 39.5i)T^{2}
83 1+(−7.29−4.21i)T+(41.5+71.8i)T2 1 + (-7.29 - 4.21i)T + (41.5 + 71.8i)T^{2}
89 1+(0.507+1.89i)T+(−77.0+44.5i)T2 1 + (0.507 + 1.89i)T + (-77.0 + 44.5i)T^{2}
97 1+(4.15−4.15i)T−97iT2 1 + (4.15 - 4.15i)T - 97iT^{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

−10.62002906821051631105634173954, −9.493393003804710855312006449517, −8.891885225729608108454327361178, −8.018376133746006848659885044394, −7.01515078120708275542238517004, −6.21575835649511817290087430557, −5.27976674613886918035892301856, −4.25538050247770726400347790500, −3.22746434661407429625462655682, −1.29569023629433021586727680618, 1.13925233186444972260261100897, 2.65940842112393880189001343408, 3.63280086091979688156531299912, 4.70725113253542381451038618250, 5.85917624308167510201974984698, 6.70409427741255474768371399092, 7.86891023372858099666431816283, 8.898259300839418803708681134807, 9.593461981277997341309709174772, 10.51251576070109077751412253749

Graph of the ZZ-function along the critical line