Properties

Label 2-55-11.2-c2-0-2
Degree 22
Conductor 5555
Sign 0.673+0.739i0.673 + 0.739i
Analytic cond. 1.498641.49864
Root an. cond. 1.224191.22419
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.40 − 1.10i)2-s + (−3.07 + 2.23i)3-s + (7.11 + 5.16i)4-s + (−0.690 − 2.12i)5-s + (12.9 − 4.19i)6-s + (4.78 − 6.58i)7-s + (−10.0 − 13.8i)8-s + (1.67 − 5.15i)9-s + 7.99i·10-s + (7.96 − 7.58i)11-s − 33.3·12-s + (20.9 + 6.79i)13-s + (−23.5 + 17.1i)14-s + (6.87 + 4.99i)15-s + (8.07 + 24.8i)16-s + (0.261 − 0.0849i)17-s + ⋯
L(s)  = 1  + (−1.70 − 0.552i)2-s + (−1.02 + 0.744i)3-s + (1.77 + 1.29i)4-s + (−0.138 − 0.425i)5-s + (2.15 − 0.699i)6-s + (0.683 − 0.940i)7-s + (−1.25 − 1.73i)8-s + (0.186 − 0.573i)9-s + 0.799i·10-s + (0.724 − 0.689i)11-s − 2.78·12-s + (1.60 + 0.522i)13-s + (−1.68 + 1.22i)14-s + (0.458 + 0.332i)15-s + (0.504 + 1.55i)16-s + (0.0153 − 0.00499i)17-s + ⋯

Functional equation

Λ(s)=(55s/2ΓC(s)L(s)=((0.673+0.739i)Λ(3s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(55s/2ΓC(s+1)L(s)=((0.673+0.739i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5115 \cdot 11
Sign: 0.673+0.739i0.673 + 0.739i
Analytic conductor: 1.498641.49864
Root analytic conductor: 1.224191.22419
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ55(46,)\chi_{55} (46, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1), 0.673+0.739i)(2,\ 55,\ (\ :1),\ 0.673 + 0.739i)

Particular Values

L(32)L(\frac{3}{2}) \approx 0.3881220.171476i0.388122 - 0.171476i
L(12)L(\frac12) \approx 0.3881220.171476i0.388122 - 0.171476i
L(2)L(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)
bad5 1+(0.690+2.12i)T 1 + (0.690 + 2.12i)T
11 1+(7.96+7.58i)T 1 + (-7.96 + 7.58i)T
good2 1+(3.40+1.10i)T+(3.23+2.35i)T2 1 + (3.40 + 1.10i)T + (3.23 + 2.35i)T^{2}
3 1+(3.072.23i)T+(2.788.55i)T2 1 + (3.07 - 2.23i)T + (2.78 - 8.55i)T^{2}
7 1+(4.78+6.58i)T+(15.146.6i)T2 1 + (-4.78 + 6.58i)T + (-15.1 - 46.6i)T^{2}
13 1+(20.96.79i)T+(136.+99.3i)T2 1 + (-20.9 - 6.79i)T + (136. + 99.3i)T^{2}
17 1+(0.261+0.0849i)T+(233.169.i)T2 1 + (-0.261 + 0.0849i)T + (233. - 169. i)T^{2}
19 1+(9.69+13.3i)T+(111.+343.i)T2 1 + (9.69 + 13.3i)T + (-111. + 343. i)T^{2}
23 11.44T+529T2 1 - 1.44T + 529T^{2}
29 1+(24.5+33.7i)T+(259.799.i)T2 1 + (-24.5 + 33.7i)T + (-259. - 799. i)T^{2}
31 1+(0.152+0.468i)T+(777.564.i)T2 1 + (-0.152 + 0.468i)T + (-777. - 564. i)T^{2}
37 1+(17.612.8i)T+(423.+1.30e3i)T2 1 + (-17.6 - 12.8i)T + (423. + 1.30e3i)T^{2}
41 1+(29.540.6i)T+(519.+1.59e3i)T2 1 + (-29.5 - 40.6i)T + (-519. + 1.59e3i)T^{2}
43 1+62.2iT1.84e3T2 1 + 62.2iT - 1.84e3T^{2}
47 1+(10.97.94i)T+(682.2.10e3i)T2 1 + (10.9 - 7.94i)T + (682. - 2.10e3i)T^{2}
53 1+(1.14+3.52i)T+(2.27e31.65e3i)T2 1 + (-1.14 + 3.52i)T + (-2.27e3 - 1.65e3i)T^{2}
59 1+(7.285.29i)T+(1.07e3+3.31e3i)T2 1 + (-7.28 - 5.29i)T + (1.07e3 + 3.31e3i)T^{2}
61 1+(78.325.4i)T+(3.01e32.18e3i)T2 1 + (78.3 - 25.4i)T + (3.01e3 - 2.18e3i)T^{2}
67 1+36.5T+4.48e3T2 1 + 36.5T + 4.48e3T^{2}
71 1+(29.691.1i)T+(4.07e3+2.96e3i)T2 1 + (-29.6 - 91.1i)T + (-4.07e3 + 2.96e3i)T^{2}
73 1+(54.0+74.3i)T+(1.64e35.06e3i)T2 1 + (-54.0 + 74.3i)T + (-1.64e3 - 5.06e3i)T^{2}
79 1+(32.6+10.6i)T+(5.04e3+3.66e3i)T2 1 + (32.6 + 10.6i)T + (5.04e3 + 3.66e3i)T^{2}
83 1+(16.95.50i)T+(5.57e34.04e3i)T2 1 + (16.9 - 5.50i)T + (5.57e3 - 4.04e3i)T^{2}
89 124.9T+7.92e3T2 1 - 24.9T + 7.92e3T^{2}
97 1+(29.089.3i)T+(7.61e35.53e3i)T2 1 + (29.0 - 89.3i)T + (-7.61e3 - 5.53e3i)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

−15.69810703907578137543848776165, −13.66894116987764217937985084180, −11.71560414705074123990276664077, −11.16205385078750279463003328156, −10.44823560711641212058195469383, −9.109331832553966072694825695874, −8.088615656008583006135202029825, −6.38893402101444723127795765848, −4.19768308509162719413946072869, −0.965149467627808304917763812519, 1.45601656647546670670135299283, 5.85631338385764292429693020540, 6.67457219709163743986137057019, 7.985209103822211211302269407054, 9.038989031574201683906556744726, 10.63931575582013890082092095722, 11.40185817597018250635761183582, 12.47074995199870384239491982284, 14.60538415604074023142527429508, 15.59219435584751185382630967461

Graph of the ZZ-function along the critical line