↑

Properties

Label 2-273-13.3-c1-0-1
Degree 22
Conductor 273273
Sign −1-1
Analytic cond. 2.179912.17991
Root an. cond. 1.476451.47645
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 + 2.05i)2-s + (0.5 + 0.866i)3-s + (−1.82 + 3.16i)4-s − 4.02·5-s + (−1.18 + 2.05i)6-s + (0.5 − 0.866i)7-s − 3.92·8-s + (−0.499 + 0.866i)9-s + (−4.78 − 8.29i)10-s + (1.63 + 2.83i)11-s − 3.65·12-s + (0.910 + 3.48i)13-s + 2.37·14-s + (−2.01 − 3.48i)15-s + (−1.01 − 1.75i)16-s + (0.188 − 0.326i)17-s + ⋯
L(s)  = 1  + (0.840 + 1.45i)2-s + (0.288 + 0.499i)3-s + (−0.912 + 1.58i)4-s − 1.80·5-s + (−0.485 + 0.840i)6-s + (0.188 − 0.327i)7-s − 1.38·8-s + (−0.166 + 0.288i)9-s + (−1.51 − 2.62i)10-s + (0.493 + 0.854i)11-s − 1.05·12-s + (0.252 + 0.967i)13-s + 0.635·14-s + (−0.520 − 0.900i)15-s + (−0.253 − 0.439i)16-s + (0.0457 − 0.0792i)17-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: −1-1
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(211,⋅)\chi_{273} (211, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), −1)(2,\ 273,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) ≈\approx −1.52306i-1.52306i
L(12)L(\frac12) ≈\approx −1.52306i-1.52306i
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
13 1+(−0.910−3.48i)T 1 + (-0.910 - 3.48i)T
good2 1+(−1.18−2.05i)T+(−1+1.73i)T2 1 + (-1.18 - 2.05i)T + (-1 + 1.73i)T^{2}
5 1+4.02T+5T2 1 + 4.02T + 5T^{2}
11 1+(−1.63−2.83i)T+(−5.5+9.52i)T2 1 + (-1.63 - 2.83i)T + (-5.5 + 9.52i)T^{2}
17 1+(−0.188+0.326i)T+(−8.5−14.7i)T2 1 + (-0.188 + 0.326i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.77+3.07i)T+(−9.5−16.4i)T2 1 + (-1.77 + 3.07i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.948+1.64i)T+(−11.5+19.9i)T2 1 + (0.948 + 1.64i)T + (-11.5 + 19.9i)T^{2}
29 1+(−4.33−7.51i)T+(−14.5+25.1i)T2 1 + (-4.33 - 7.51i)T + (-14.5 + 25.1i)T^{2}
31 1−6.33T+31T2 1 - 6.33T + 31T^{2}
37 1+(2.01+3.48i)T+(−18.5+32.0i)T2 1 + (2.01 + 3.48i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.75+6.50i)T+(−20.5+35.5i)T2 1 + (3.75 + 6.50i)T + (-20.5 + 35.5i)T^{2}
43 1+(−4.32+7.49i)T+(−21.5−37.2i)T2 1 + (-4.32 + 7.49i)T + (-21.5 - 37.2i)T^{2}
47 1+12.1T+47T2 1 + 12.1T + 47T^{2}
53 1−7.75T+53T2 1 - 7.75T + 53T^{2}
59 1+(−1.05+1.82i)T+(−29.5−51.0i)T2 1 + (-1.05 + 1.82i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.87−6.71i)T+(−30.5−52.8i)T2 1 + (3.87 - 6.71i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.79+4.83i)T+(−33.5+58.0i)T2 1 + (2.79 + 4.83i)T + (-33.5 + 58.0i)T^{2}
71 1+(1.99−3.45i)T+(−35.5−61.4i)T2 1 + (1.99 - 3.45i)T + (-35.5 - 61.4i)T^{2}
73 1−7.50T+73T2 1 - 7.50T + 73T^{2}
79 1+1.16T+79T2 1 + 1.16T + 79T^{2}
83 1−15.1T+83T2 1 - 15.1T + 83T^{2}
89 1+(3.17+5.50i)T+(−44.5+77.0i)T2 1 + (3.17 + 5.50i)T + (-44.5 + 77.0i)T^{2}
97 1+(2.48−4.30i)T+(−48.5−84.0i)T2 1 + (2.48 - 4.30i)T + (-48.5 - 84.0i)T^{2}
show more
show less
   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

−12.37819790407729118517491767116, −11.77110940733718819290160107377, −10.60700279128365456105543936501, −8.984997697948884105747703920331, −8.260934037346162988440165959347, −7.23151300441027167955305607904, −6.78588432394752831969086912991, −4.93980089316024543068307165808, −4.32797491109275325481141649353, −3.52235045553583954417208475145, 0.970504142698149759165400378650, 2.96596235019041213893525120197, 3.64462008834228693150872169353, 4.74678904748333587912981009639, 6.18865687689488090115080754141, 7.87757728860861122416237447691, 8.357545273876155913220685641963, 9.872929748328341556755349805056, 11.00470887568194829066776455476, 11.76544702288331593818059683636

Graph of the ZZ-function along the critical line