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Properties

Label 2-273-91.16-c1-0-15
Degree 22
Conductor 273273
Sign 0.946+0.323i0.946 + 0.323i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.05·2-s + (0.5 − 0.866i)3-s + 2.22·4-s + (−0.274 + 0.475i)5-s + (1.02 − 1.77i)6-s + (2.59 − 0.527i)7-s + 0.456·8-s + (−0.499 − 0.866i)9-s + (−0.564 + 0.977i)10-s + (−2.34 + 4.06i)11-s + (1.11 − 1.92i)12-s + (−0.663 − 3.54i)13-s + (5.32 − 1.08i)14-s + (0.274 + 0.475i)15-s − 3.50·16-s − 0.603·17-s + ⋯
L(s)  = 1  + 1.45·2-s + (0.288 − 0.499i)3-s + 1.11·4-s + (−0.122 + 0.212i)5-s + (0.419 − 0.726i)6-s + (0.979 − 0.199i)7-s + 0.161·8-s + (−0.166 − 0.288i)9-s + (−0.178 + 0.309i)10-s + (−0.708 + 1.22i)11-s + (0.320 − 0.555i)12-s + (−0.184 − 0.982i)13-s + (1.42 − 0.289i)14-s + (0.0709 + 0.122i)15-s − 0.876·16-s − 0.146·17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.946+0.323i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((0.946+0.323i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.946+0.323i0.946 + 0.323i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(16,⋅)\chi_{273} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.946+0.323i)(2,\ 273,\ (\ :1/2),\ 0.946 + 0.323i)

Particular Values

L(1)L(1) ≈\approx 2.70635−0.450350i2.70635 - 0.450350i
L(12)L(\frac12) ≈\approx 2.70635−0.450350i2.70635 - 0.450350i
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+(−2.59+0.527i)T 1 + (-2.59 + 0.527i)T
13 1+(0.663+3.54i)T 1 + (0.663 + 3.54i)T
good2 1−2.05T+2T2 1 - 2.05T + 2T^{2}
5 1+(0.274−0.475i)T+(−2.5−4.33i)T2 1 + (0.274 - 0.475i)T + (-2.5 - 4.33i)T^{2}
11 1+(2.34−4.06i)T+(−5.5−9.52i)T2 1 + (2.34 - 4.06i)T + (-5.5 - 9.52i)T^{2}
17 1+0.603T+17T2 1 + 0.603T + 17T^{2}
19 1+(0.280+0.485i)T+(−9.5+16.4i)T2 1 + (0.280 + 0.485i)T + (-9.5 + 16.4i)T^{2}
23 1−0.376T+23T2 1 - 0.376T + 23T^{2}
29 1+(−2.09−3.62i)T+(−14.5+25.1i)T2 1 + (-2.09 - 3.62i)T + (-14.5 + 25.1i)T^{2}
31 1+(0.577+0.999i)T+(−15.5+26.8i)T2 1 + (0.577 + 0.999i)T + (-15.5 + 26.8i)T^{2}
37 1+8.80T+37T2 1 + 8.80T + 37T^{2}
41 1+(3.96+6.85i)T+(−20.5+35.5i)T2 1 + (3.96 + 6.85i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.747−1.29i)T+(−21.5−37.2i)T2 1 + (0.747 - 1.29i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.09−1.90i)T+(−23.5−40.7i)T2 1 + (1.09 - 1.90i)T + (-23.5 - 40.7i)T^{2}
53 1+(−4.52−7.83i)T+(−26.5+45.8i)T2 1 + (-4.52 - 7.83i)T + (-26.5 + 45.8i)T^{2}
59 1−8.53T+59T2 1 - 8.53T + 59T^{2}
61 1+(3.71+6.42i)T+(−30.5+52.8i)T2 1 + (3.71 + 6.42i)T + (-30.5 + 52.8i)T^{2}
67 1+(−4.79+8.31i)T+(−33.5−58.0i)T2 1 + (-4.79 + 8.31i)T + (-33.5 - 58.0i)T^{2}
71 1+(−2.88+5.00i)T+(−35.5−61.4i)T2 1 + (-2.88 + 5.00i)T + (-35.5 - 61.4i)T^{2}
73 1+(−7.24−12.5i)T+(−36.5+63.2i)T2 1 + (-7.24 - 12.5i)T + (-36.5 + 63.2i)T^{2}
79 1+(−7.31+12.6i)T+(−39.5−68.4i)T2 1 + (-7.31 + 12.6i)T + (-39.5 - 68.4i)T^{2}
83 1+14.8T+83T2 1 + 14.8T + 83T^{2}
89 1−9.18T+89T2 1 - 9.18T + 89T^{2}
97 1+(−3.15+5.45i)T+(−48.5−84.0i)T2 1 + (-3.15 + 5.45i)T + (-48.5 - 84.0i)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

−12.29269105365733184428775686212, −11.20699992775944011521677019642, −10.29537891577595015177998481402, −8.778486359152685546173271165635, −7.60154904970995619121038999201, −6.90354253933115546225911032161, −5.41724387305438855431616730182, −4.75966617864777879363003197982, −3.38671145260066128958807052597, −2.12676436934481645113374256720, 2.42021904308716418993873679538, 3.72177639620086344223807054618, 4.74811778516330195704552670470, 5.44814630057996683606345906098, 6.67322512150191429225763125394, 8.206855217722597339604626291692, 8.872398993563301408469863375665, 10.35862624388132104106164996173, 11.39450179686526896727301432861, 11.90969090053810999398394895537

Graph of the ZZ-function along the critical line