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Properties

Label 2-538-269.21-c1-0-13
Degree 22
Conductor 538538
Sign −0.117−0.993i-0.117 - 0.993i
Analytic cond. 4.295954.29595
Root an. cond. 2.072662.07266
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  + (0.0702 + 0.997i)2-s + (1.38 + 0.262i)3-s + (−0.990 + 0.140i)4-s + (1.40 + 2.62i)5-s + (−0.164 + 1.39i)6-s + (3.20 + 0.764i)7-s + (−0.209 − 0.977i)8-s + (−0.951 − 0.374i)9-s + (−2.51 + 1.58i)10-s + (1.81 − 0.918i)11-s + (−1.40 − 0.0659i)12-s + (−0.642 + 0.405i)13-s + (−0.537 + 3.24i)14-s + (1.25 + 3.99i)15-s + (0.960 − 0.277i)16-s + (−1.43 − 0.724i)17-s + ⋯
L(s)  = 1  + (0.0496 + 0.705i)2-s + (0.797 + 0.151i)3-s + (−0.495 + 0.0701i)4-s + (0.629 + 1.17i)5-s + (−0.0671 + 0.570i)6-s + (1.20 + 0.288i)7-s + (−0.0740 − 0.345i)8-s + (−0.317 − 0.124i)9-s + (−0.796 + 0.502i)10-s + (0.546 − 0.277i)11-s + (−0.405 − 0.0190i)12-s + (−0.178 + 0.112i)13-s + (−0.143 + 0.867i)14-s + (0.324 + 1.03i)15-s + (0.240 − 0.0694i)16-s + (−0.347 − 0.175i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((−0.117−0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((−0.117−0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −0.117−0.993i-0.117 - 0.993i
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ538(21,⋅)\chi_{538} (21, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), −0.117−0.993i)(2,\ 538,\ (\ :1/2),\ -0.117 - 0.993i)

Particular Values

L(1)L(1) ≈\approx 1.42739+1.60638i1.42739 + 1.60638i
L(12)L(\frac12) ≈\approx 1.42739+1.60638i1.42739 + 1.60638i
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)
bad2 1+(−0.0702−0.997i)T 1 + (-0.0702 - 0.997i)T
269 1+(−10.0−12.9i)T 1 + (-10.0 - 12.9i)T
good3 1+(−1.38−0.262i)T+(2.79+1.09i)T2 1 + (-1.38 - 0.262i)T + (2.79 + 1.09i)T^{2}
5 1+(−1.40−2.62i)T+(−2.76+4.16i)T2 1 + (-1.40 - 2.62i)T + (-2.76 + 4.16i)T^{2}
7 1+(−3.20−0.764i)T+(6.24+3.16i)T2 1 + (-3.20 - 0.764i)T + (6.24 + 3.16i)T^{2}
11 1+(−1.81+0.918i)T+(6.50−8.86i)T2 1 + (-1.81 + 0.918i)T + (6.50 - 8.86i)T^{2}
13 1+(0.642−0.405i)T+(5.60−11.7i)T2 1 + (0.642 - 0.405i)T + (5.60 - 11.7i)T^{2}
17 1+(1.43+0.724i)T+(10.0+13.7i)T2 1 + (1.43 + 0.724i)T + (10.0 + 13.7i)T^{2}
19 1+(−0.000188+0.00804i)T+(−18.9−0.890i)T2 1 + (-0.000188 + 0.00804i)T + (-18.9 - 0.890i)T^{2}
23 1+(−2.02−0.384i)T+(21.4+8.42i)T2 1 + (-2.02 - 0.384i)T + (21.4 + 8.42i)T^{2}
29 1+(5.63+8.49i)T+(−11.2+26.7i)T2 1 + (5.63 + 8.49i)T + (-11.2 + 26.7i)T^{2}
31 1+(0.417−0.568i)T+(−9.30−29.5i)T2 1 + (0.417 - 0.568i)T + (-9.30 - 29.5i)T^{2}
37 1+(0.00525−0.00538i)T+(−0.867−36.9i)T2 1 + (0.00525 - 0.00538i)T + (-0.867 - 36.9i)T^{2}
41 1+(0.556−7.90i)T+(−40.5−5.74i)T2 1 + (0.556 - 7.90i)T + (-40.5 - 5.74i)T^{2}
43 1+(−3.96−5.97i)T+(−16.6+39.6i)T2 1 + (-3.96 - 5.97i)T + (-16.6 + 39.6i)T^{2}
47 1+(3.57+4.42i)T+(−9.84+45.9i)T2 1 + (3.57 + 4.42i)T + (-9.84 + 45.9i)T^{2}
53 1+(1.30−3.56i)T+(−40.4−34.2i)T2 1 + (1.30 - 3.56i)T + (-40.4 - 34.2i)T^{2}
59 1+(1.38−0.195i)T+(56.6−16.3i)T2 1 + (1.38 - 0.195i)T + (56.6 - 16.3i)T^{2}
61 1+(−4.35+4.90i)T+(−7.13−60.5i)T2 1 + (-4.35 + 4.90i)T + (-7.13 - 60.5i)T^{2}
67 1+(−1.68−0.239i)T+(64.3+18.6i)T2 1 + (-1.68 - 0.239i)T + (64.3 + 18.6i)T^{2}
71 1+(0.507+3.06i)T+(−67.2+22.8i)T2 1 + (0.507 + 3.06i)T + (-67.2 + 22.8i)T^{2}
73 1+(−6.24−2.12i)T+(57.8+44.5i)T2 1 + (-6.24 - 2.12i)T + (57.8 + 44.5i)T^{2}
79 1+(1.53+2.56i)T+(−37.3+69.6i)T2 1 + (1.53 + 2.56i)T + (-37.3 + 69.6i)T^{2}
83 1+(4.39+4.09i)T+(5.83+82.7i)T2 1 + (4.39 + 4.09i)T + (5.83 + 82.7i)T^{2}
89 1+(4.10+15.5i)T+(−77.4+43.8i)T2 1 + (4.10 + 15.5i)T + (-77.4 + 43.8i)T^{2}
97 1+(−4.08−4.59i)T+(−11.3+96.3i)T2 1 + (-4.08 - 4.59i)T + (-11.3 + 96.3i)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

−11.16862535747160176696894877305, −9.903386559097434671696854051679, −9.193040632250956574715031580772, −8.306705659626865049573182851059, −7.56535781325283350096359554195, −6.48950631284623813154433048925, −5.70949693445334167430551315821, −4.45200199904331949391475516914, −3.18647720189889933263904683739, −2.08304997783037481585523018891, 1.36316151361428011588022167869, 2.19895684125340067869421056243, 3.74443243090321511952502759907, 4.86471782466749605644576454313, 5.52077347757194306398988870546, 7.24103245705967536769882407170, 8.288751467066349220807838288015, 8.887083779669226731874795276456, 9.457556998375003008637786351962, 10.69762758077902887670899921929

Graph of the ZZ-function along the critical line