Properties

Label 2-273-91.59-c1-0-8
Degree 22
Conductor 273273
Sign 0.1100.993i-0.110 - 0.993i
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  + (1.85 + 1.85i)2-s + (0.866 − 0.5i)3-s + 4.91i·4-s + (0.502 + 0.134i)5-s + (2.53 + 0.680i)6-s + (−1.89 − 1.84i)7-s + (−5.41 + 5.41i)8-s + (0.499 − 0.866i)9-s + (0.684 + 1.18i)10-s + (−1.93 − 0.517i)11-s + (2.45 + 4.25i)12-s + (2.40 + 2.68i)13-s + (−0.0921 − 6.95i)14-s + (0.502 − 0.134i)15-s − 10.3·16-s + 5.09·17-s + ⋯
L(s)  = 1  + (1.31 + 1.31i)2-s + (0.499 − 0.288i)3-s + 2.45i·4-s + (0.224 + 0.0602i)5-s + (1.03 + 0.277i)6-s + (−0.716 − 0.697i)7-s + (−1.91 + 1.91i)8-s + (0.166 − 0.288i)9-s + (0.216 + 0.374i)10-s + (−0.582 − 0.156i)11-s + (0.709 + 1.22i)12-s + (0.667 + 0.744i)13-s + (−0.0246 − 1.85i)14-s + (0.129 − 0.0347i)15-s − 2.57·16-s + 1.23·17-s + ⋯

Functional equation

Λ(s)=(273s/2ΓC(s)L(s)=((0.1100.993i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2ΓC(s+1/2)L(s)=((0.1100.993i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    37133 \cdot 7 \cdot 13
Sign: 0.1100.993i-0.110 - 0.993i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(241,)\chi_{273} (241, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.1100.993i)(2,\ 273,\ (\ :1/2),\ -0.110 - 0.993i)

Particular Values

L(1)L(1) \approx 1.76119+1.96876i1.76119 + 1.96876i
L(12)L(\frac12) \approx 1.76119+1.96876i1.76119 + 1.96876i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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+(1.89+1.84i)T 1 + (1.89 + 1.84i)T
13 1+(2.402.68i)T 1 + (-2.40 - 2.68i)T
good2 1+(1.851.85i)T+2iT2 1 + (-1.85 - 1.85i)T + 2iT^{2}
5 1+(0.5020.134i)T+(4.33+2.5i)T2 1 + (-0.502 - 0.134i)T + (4.33 + 2.5i)T^{2}
11 1+(1.93+0.517i)T+(9.52+5.5i)T2 1 + (1.93 + 0.517i)T + (9.52 + 5.5i)T^{2}
17 15.09T+17T2 1 - 5.09T + 17T^{2}
19 1+(1.86+6.96i)T+(16.4+9.5i)T2 1 + (1.86 + 6.96i)T + (-16.4 + 9.5i)T^{2}
23 1+4.78iT23T2 1 + 4.78iT - 23T^{2}
29 1+(1.352.34i)T+(14.525.1i)T2 1 + (1.35 - 2.34i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.8673.23i)T+(26.8+15.5i)T2 1 + (-0.867 - 3.23i)T + (-26.8 + 15.5i)T^{2}
37 1+(5.255.25i)T37iT2 1 + (5.25 - 5.25i)T - 37iT^{2}
41 1+(0.9383.50i)T+(35.5+20.5i)T2 1 + (-0.938 - 3.50i)T + (-35.5 + 20.5i)T^{2}
43 1+(3.622.09i)T+(21.537.2i)T2 1 + (3.62 - 2.09i)T + (21.5 - 37.2i)T^{2}
47 1+(0.0215+0.0803i)T+(40.723.5i)T2 1 + (-0.0215 + 0.0803i)T + (-40.7 - 23.5i)T^{2}
53 1+(6.85+11.8i)T+(26.545.8i)T2 1 + (-6.85 + 11.8i)T + (-26.5 - 45.8i)T^{2}
59 1+(3.91+3.91i)T+59iT2 1 + (3.91 + 3.91i)T + 59iT^{2}
61 1+(0.652+0.376i)T+(30.5+52.8i)T2 1 + (0.652 + 0.376i)T + (30.5 + 52.8i)T^{2}
67 1+(2.8110.4i)T+(58.033.5i)T2 1 + (2.81 - 10.4i)T + (-58.0 - 33.5i)T^{2}
71 1+(1.335.00i)T+(61.435.5i)T2 1 + (1.33 - 5.00i)T + (-61.4 - 35.5i)T^{2}
73 1+(2.05+0.551i)T+(63.236.5i)T2 1 + (-2.05 + 0.551i)T + (63.2 - 36.5i)T^{2}
79 1+(6.1710.6i)T+(39.5+68.4i)T2 1 + (-6.17 - 10.6i)T + (-39.5 + 68.4i)T^{2}
83 1+(7.09+7.09i)T83iT2 1 + (-7.09 + 7.09i)T - 83iT^{2}
89 1+(9.64+9.64i)T+89iT2 1 + (9.64 + 9.64i)T + 89iT^{2}
97 1+(3.28+0.880i)T+(84.0+48.5i)T2 1 + (3.28 + 0.880i)T + (84.0 + 48.5i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−12.74485551592370900194132972208, −11.64141264876491979224224495487, −10.21086216823439901329881835299, −8.842943775132587375975980048361, −7.947526223021500531702630479489, −6.88673165287671898961379998529, −6.40085617293384239038192517744, −5.08793620619979569467294625834, −3.92443388983074872860193357384, −2.90080146957259594864366034840, 1.87847759347150946885284929564, 3.17328488885341760071016080032, 3.85747761596617113510230258920, 5.54239541314921387225821652987, 5.82991547246538118793507007726, 7.84497591658453515952439515624, 9.303050374738053039367055153789, 10.06592534349962668920262965189, 10.69882510536668582962052064434, 12.00642904156962593914008830341

Graph of the ZZ-function along the critical line