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Properties

Label 2-273-273.107-c0-0-1
Degree 22
Conductor 273273
Sign 0.190−0.981i0.190 - 0.981i
Analytic cond. 0.1362440.136244
Root an. cond. 0.3691130.369113
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s − 13-s + (0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s − 16-s − i·17-s + (−0.866 + 0.499i)18-s − 0.999i·21-s + ⋯
L(s)  = 1  + i·2-s + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s − 13-s + (0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s − 16-s − i·17-s + (−0.866 + 0.499i)18-s − 0.999i·21-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.190−0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s) L(s)=((0.190−0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.190−0.981i0.190 - 0.981i
Analytic conductor: 0.1362440.136244
Root analytic conductor: 0.3691130.369113
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ273(107,⋅)\chi_{273} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :0), 0.190−0.981i)(2,\ 273,\ (\ :0),\ 0.190 - 0.981i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.89024854010.8902485401
L(12)L(\frac12) ≈\approx 0.89024854010.8902485401
L(1)L(1) 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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
7 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1+T 1 + T
good2 1−iT−T2 1 - iT - T^{2}
5 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+iT−T2 1 + iT - T^{2}
19 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
23 1+iT−T2 1 + iT - T^{2}
29 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
31 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1−T+T2 1 - T + T^{2}
41 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
43 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
59 1−iT−T2 1 - iT - T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
73 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1−iT−T2 1 - iT - T^{2}
97 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)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

−12.45839087364309845829082429501, −11.34893602192075396821191455166, −10.27090992328387082700100767647, −9.261984649232099377938399527062, −8.231170295604727435905978789256, −7.51828293431547257653370785905, −6.79630155492941171133576287977, −5.07790150804553031667076782138, −4.21133916261879778572438295852, −2.75991142853910062421931303486, 2.09368771117900151686760285036, 3.08829971249185133363349303392, 3.99770012676609839692866788023, 6.11384775433892806220920232724, 7.22864699467596249750084034109, 8.007607814910001464625463047681, 9.337069498874731396699529351394, 9.946932265950353662835940003386, 11.28242428730413389059665891949, 11.90363920371632228224238796993

Graph of the ZZ-function along the critical line