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Properties

Label 2-273-91.54-c1-0-0
Degree 22
Conductor 273273
Sign −0.431+0.902i-0.431 + 0.902i
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  + (−1.22 + 1.22i)2-s + (0.866 + 0.5i)3-s − 1.02i·4-s + (−1.95 + 0.524i)5-s + (−1.67 + 0.449i)6-s + (−1.82 − 1.92i)7-s + (−1.20 − 1.20i)8-s + (0.499 + 0.866i)9-s + (1.76 − 3.05i)10-s + (−3.50 + 0.938i)11-s + (0.511 − 0.885i)12-s + (−1.78 − 3.13i)13-s + (4.59 + 0.123i)14-s + (−1.95 − 0.524i)15-s + 4.99·16-s − 1.50·17-s + ⋯
L(s)  = 1  + (−0.869 + 0.869i)2-s + (0.499 + 0.288i)3-s − 0.511i·4-s + (−0.875 + 0.234i)5-s + (−0.685 + 0.183i)6-s + (−0.687 − 0.725i)7-s + (−0.424 − 0.424i)8-s + (0.166 + 0.288i)9-s + (0.556 − 0.964i)10-s + (−1.05 + 0.282i)11-s + (0.147 − 0.255i)12-s + (−0.496 − 0.868i)13-s + (1.22 + 0.0328i)14-s + (−0.505 − 0.135i)15-s + 1.24·16-s − 0.365·17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((−0.431+0.902i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((−0.431+0.902i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: −0.431+0.902i-0.431 + 0.902i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(145,⋅)\chi_{273} (145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), −0.431+0.902i)(2,\ 273,\ (\ :1/2),\ -0.431 + 0.902i)

Particular Values

L(1)L(1) ≈\approx 0.0346451−0.0549476i0.0346451 - 0.0549476i
L(12)L(\frac12) ≈\approx 0.0346451−0.0549476i0.0346451 - 0.0549476i
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
7 1+(1.82+1.92i)T 1 + (1.82 + 1.92i)T
13 1+(1.78+3.13i)T 1 + (1.78 + 3.13i)T
good2 1+(1.22−1.22i)T−2iT2 1 + (1.22 - 1.22i)T - 2iT^{2}
5 1+(1.95−0.524i)T+(4.33−2.5i)T2 1 + (1.95 - 0.524i)T + (4.33 - 2.5i)T^{2}
11 1+(3.50−0.938i)T+(9.52−5.5i)T2 1 + (3.50 - 0.938i)T + (9.52 - 5.5i)T^{2}
17 1+1.50T+17T2 1 + 1.50T + 17T^{2}
19 1+(1.07−3.99i)T+(−16.4−9.5i)T2 1 + (1.07 - 3.99i)T + (-16.4 - 9.5i)T^{2}
23 1−3.71iT−23T2 1 - 3.71iT - 23T^{2}
29 1+(1.84+3.19i)T+(−14.5+25.1i)T2 1 + (1.84 + 3.19i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.89+7.05i)T+(−26.8−15.5i)T2 1 + (-1.89 + 7.05i)T + (-26.8 - 15.5i)T^{2}
37 1+(−1.85−1.85i)T+37iT2 1 + (-1.85 - 1.85i)T + 37iT^{2}
41 1+(1.33−4.97i)T+(−35.5−20.5i)T2 1 + (1.33 - 4.97i)T + (-35.5 - 20.5i)T^{2}
43 1+(4.51+2.60i)T+(21.5+37.2i)T2 1 + (4.51 + 2.60i)T + (21.5 + 37.2i)T^{2}
47 1+(−0.0684−0.255i)T+(−40.7+23.5i)T2 1 + (-0.0684 - 0.255i)T + (-40.7 + 23.5i)T^{2}
53 1+(2.63+4.57i)T+(−26.5+45.8i)T2 1 + (2.63 + 4.57i)T + (-26.5 + 45.8i)T^{2}
59 1+(−0.912+0.912i)T−59iT2 1 + (-0.912 + 0.912i)T - 59iT^{2}
61 1+(9.19−5.30i)T+(30.5−52.8i)T2 1 + (9.19 - 5.30i)T + (30.5 - 52.8i)T^{2}
67 1+(−3.47−12.9i)T+(−58.0+33.5i)T2 1 + (-3.47 - 12.9i)T + (-58.0 + 33.5i)T^{2}
71 1+(−3.11−11.6i)T+(−61.4+35.5i)T2 1 + (-3.11 - 11.6i)T + (-61.4 + 35.5i)T^{2}
73 1+(2.77+0.744i)T+(63.2+36.5i)T2 1 + (2.77 + 0.744i)T + (63.2 + 36.5i)T^{2}
79 1+(8.09−14.0i)T+(−39.5−68.4i)T2 1 + (8.09 - 14.0i)T + (-39.5 - 68.4i)T^{2}
83 1+(10.3+10.3i)T+83iT2 1 + (10.3 + 10.3i)T + 83iT^{2}
89 1+(−7.11+7.11i)T−89iT2 1 + (-7.11 + 7.11i)T - 89iT^{2}
97 1+(0.645−0.173i)T+(84.0−48.5i)T2 1 + (0.645 - 0.173i)T + (84.0 - 48.5i)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

−12.72060189273642917528031489812, −11.42357186713349067018486012849, −10.07758139820559528446760142984, −9.831019474630209336244212171096, −8.362141500077479040221590080521, −7.73495735351038669540644770390, −7.17681221828975507117732216807, −5.80819970956113551345348299188, −4.11438598964697923485524792060, −3.04585961357718623292603609099, 0.05770882027911960509757836864, 2.21913034365176255783469268904, 3.20777279438795672050287961178, 4.89707505875763272288233564025, 6.46420593421485022281886651354, 7.72553940429404996031580806661, 8.706610520458006179078041319459, 9.179362295874296594759800535252, 10.30589403627070328396230704067, 11.20682414600050673309404858930

Graph of the ZZ-function along the critical line