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Properties

Label 2-273-91.9-c1-0-5
Degree 22
Conductor 273273
Sign 0.982+0.183i0.982 + 0.183i
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  + (−0.613 − 1.06i)2-s + 3-s + (0.247 − 0.429i)4-s + (−2.10 + 3.64i)5-s + (−0.613 − 1.06i)6-s + (2.23 + 1.41i)7-s − 3.06·8-s + 9-s + 5.16·10-s + 5.52·11-s + (0.247 − 0.429i)12-s + (3.59 − 0.226i)13-s + (0.139 − 3.24i)14-s + (−2.10 + 3.64i)15-s + (1.38 + 2.39i)16-s + (−0.0891 + 0.154i)17-s + ⋯
L(s)  = 1  + (−0.433 − 0.751i)2-s + 0.577·3-s + (0.123 − 0.214i)4-s + (−0.942 + 1.63i)5-s + (−0.250 − 0.433i)6-s + (0.843 + 0.536i)7-s − 1.08·8-s + 0.333·9-s + 1.63·10-s + 1.66·11-s + (0.0715 − 0.123i)12-s + (0.998 − 0.0627i)13-s + (0.0372 − 0.866i)14-s + (−0.543 + 0.942i)15-s + (0.345 + 0.598i)16-s + (−0.0216 + 0.0374i)17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.982+0.183i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((0.982+0.183i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.982+0.183i0.982 + 0.183i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(100,⋅)\chi_{273} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.982+0.183i)(2,\ 273,\ (\ :1/2),\ 0.982 + 0.183i)

Particular Values

L(1)L(1) ≈\approx 1.25143−0.116095i1.25143 - 0.116095i
L(12)L(\frac12) ≈\approx 1.25143−0.116095i1.25143 - 0.116095i
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−T 1 - T
7 1+(−2.23−1.41i)T 1 + (-2.23 - 1.41i)T
13 1+(−3.59+0.226i)T 1 + (-3.59 + 0.226i)T
good2 1+(0.613+1.06i)T+(−1+1.73i)T2 1 + (0.613 + 1.06i)T + (-1 + 1.73i)T^{2}
5 1+(2.10−3.64i)T+(−2.5−4.33i)T2 1 + (2.10 - 3.64i)T + (-2.5 - 4.33i)T^{2}
11 1−5.52T+11T2 1 - 5.52T + 11T^{2}
17 1+(0.0891−0.154i)T+(−8.5−14.7i)T2 1 + (0.0891 - 0.154i)T + (-8.5 - 14.7i)T^{2}
19 1+4.51T+19T2 1 + 4.51T + 19T^{2}
23 1+(0.543+0.941i)T+(−11.5+19.9i)T2 1 + (0.543 + 0.941i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.0731−0.126i)T+(−14.5−25.1i)T2 1 + (0.0731 - 0.126i)T + (-14.5 - 25.1i)T^{2}
31 1+(−4.19−7.26i)T+(−15.5+26.8i)T2 1 + (-4.19 - 7.26i)T + (-15.5 + 26.8i)T^{2}
37 1+(2.15+3.73i)T+(−18.5+32.0i)T2 1 + (2.15 + 3.73i)T + (-18.5 + 32.0i)T^{2}
41 1+(−0.782+1.35i)T+(−20.5−35.5i)T2 1 + (-0.782 + 1.35i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.66+2.88i)T+(−21.5+37.2i)T2 1 + (1.66 + 2.88i)T + (-21.5 + 37.2i)T^{2}
47 1+(−0.636+1.10i)T+(−23.5−40.7i)T2 1 + (-0.636 + 1.10i)T + (-23.5 - 40.7i)T^{2}
53 1+(3.93+6.82i)T+(−26.5+45.8i)T2 1 + (3.93 + 6.82i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1.01+1.75i)T+(−29.5−51.0i)T2 1 + (-1.01 + 1.75i)T + (-29.5 - 51.0i)T^{2}
61 1+3.76T+61T2 1 + 3.76T + 61T^{2}
67 1+0.307T+67T2 1 + 0.307T + 67T^{2}
71 1+(1.62+2.82i)T+(−35.5+61.4i)T2 1 + (1.62 + 2.82i)T + (-35.5 + 61.4i)T^{2}
73 1+(3.53+6.11i)T+(−36.5+63.2i)T2 1 + (3.53 + 6.11i)T + (-36.5 + 63.2i)T^{2}
79 1+(2.30−3.99i)T+(−39.5−68.4i)T2 1 + (2.30 - 3.99i)T + (-39.5 - 68.4i)T^{2}
83 1+8.85T+83T2 1 + 8.85T + 83T^{2}
89 1+(6.59+11.4i)T+(−44.5+77.0i)T2 1 + (6.59 + 11.4i)T + (-44.5 + 77.0i)T^{2}
97 1+(−1.17−2.03i)T+(−48.5+84.0i)T2 1 + (-1.17 - 2.03i)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

−11.59342770516434941923286366316, −10.97282099454650590207986818185, −10.27957640187194709467388994767, −8.957970132498888324321070642767, −8.307175235249411823043356075961, −6.91629697796176645813457376763, −6.21128983638838104305515465233, −4.05688292190028729707579040339, −3.09751984824603103040968409916, −1.79727083683669845454963174553, 1.27975158432179767045686574033, 3.82229366897289201960633030166, 4.44844956686828404759991312706, 6.15546047892485104048789334065, 7.36816411265250573399691343374, 8.302988401372126269590144192200, 8.607528146684777434517997648922, 9.461974453527608186168153958975, 11.31129554090089753534151449298, 11.87624740566773253148987089273

Graph of the ZZ-function along the critical line