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Properties

Label 2-538-269.103-c1-0-11
Degree 22
Conductor 538538
Sign 0.999−0.0198i0.999 - 0.0198i
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  + (−0.366 − 0.930i)2-s + (0.0594 + 1.26i)3-s + (−0.731 + 0.681i)4-s + (3.62 + 2.05i)5-s + (1.15 − 0.519i)6-s + (−2.31 − 2.06i)7-s + (0.902 + 0.430i)8-s + (1.38 − 0.130i)9-s + (0.583 − 4.12i)10-s + (0.568 − 4.82i)11-s + (−0.907 − 0.886i)12-s + (3.30 + 0.467i)13-s + (−1.06 + 2.91i)14-s + (−2.38 + 4.71i)15-s + (0.0702 − 0.997i)16-s + (0.156 − 0.0184i)17-s + ⋯
L(s)  = 1  + (−0.259 − 0.657i)2-s + (0.0343 + 0.731i)3-s + (−0.365 + 0.340i)4-s + (1.61 + 0.918i)5-s + (0.472 − 0.212i)6-s + (−0.876 − 0.779i)7-s + (0.319 + 0.152i)8-s + (0.461 − 0.0433i)9-s + (0.184 − 1.30i)10-s + (0.171 − 1.45i)11-s + (−0.262 − 0.255i)12-s + (0.916 + 0.129i)13-s + (−0.285 + 0.778i)14-s + (−0.616 + 1.21i)15-s + (0.0175 − 0.249i)16-s + (0.0379 − 0.00447i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((0.999−0.0198i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0198i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((0.999−0.0198i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0198i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.999−0.0198i0.999 - 0.0198i
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ538(103,⋅)\chi_{538} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), 0.999−0.0198i)(2,\ 538,\ (\ :1/2),\ 0.999 - 0.0198i)

Particular Values

L(1)L(1) ≈\approx 1.61929+0.0160836i1.61929 + 0.0160836i
L(12)L(\frac12) ≈\approx 1.61929+0.0160836i1.61929 + 0.0160836i
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+(0.366+0.930i)T 1 + (0.366 + 0.930i)T
269 1+(−16.2−2.14i)T 1 + (-16.2 - 2.14i)T
good3 1+(−0.0594−1.26i)T+(−2.98+0.280i)T2 1 + (-0.0594 - 1.26i)T + (-2.98 + 0.280i)T^{2}
5 1+(−3.62−2.05i)T+(2.56+4.29i)T2 1 + (-3.62 - 2.05i)T + (2.56 + 4.29i)T^{2}
7 1+(2.31+2.06i)T+(0.818+6.95i)T2 1 + (2.31 + 2.06i)T + (0.818 + 6.95i)T^{2}
11 1+(−0.568+4.82i)T+(−10.6−2.55i)T2 1 + (-0.568 + 4.82i)T + (-10.6 - 2.55i)T^{2}
13 1+(−3.30−0.467i)T+(12.4+3.60i)T2 1 + (-3.30 - 0.467i)T + (12.4 + 3.60i)T^{2}
17 1+(−0.156+0.0184i)T+(16.5−3.94i)T2 1 + (-0.156 + 0.0184i)T + (16.5 - 3.94i)T^{2}
19 1+(3.73−1.57i)T+(13.2−13.5i)T2 1 + (3.73 - 1.57i)T + (13.2 - 13.5i)T^{2}
23 1+(−6.62+0.310i)T+(22.8−2.15i)T2 1 + (-6.62 + 0.310i)T + (22.8 - 2.15i)T^{2}
29 1+(−4.30−2.57i)T+(13.7+25.5i)T2 1 + (-4.30 - 2.57i)T + (13.7 + 25.5i)T^{2}
31 1+(1.92−8.06i)T+(−27.6−14.0i)T2 1 + (1.92 - 8.06i)T + (-27.6 - 14.0i)T^{2}
37 1+(5.42+8.16i)T+(−14.3+34.0i)T2 1 + (5.42 + 8.16i)T + (-14.3 + 34.0i)T^{2}
41 1+(3.01+1.18i)T+(29.9+27.9i)T2 1 + (3.01 + 1.18i)T + (29.9 + 27.9i)T^{2}
43 1+(4.08−6.82i)T+(−20.3−37.8i)T2 1 + (4.08 - 6.82i)T + (-20.3 - 37.8i)T^{2}
47 1+(2.00+1.26i)T+(20.2+42.4i)T2 1 + (2.00 + 1.26i)T + (20.2 + 42.4i)T^{2}
53 1+(0.340−1.08i)T+(−43.4−30.3i)T2 1 + (0.340 - 1.08i)T + (-43.4 - 30.3i)T^{2}
59 1+(5.07+5.44i)T+(−4.14+58.8i)T2 1 + (5.07 + 5.44i)T + (-4.14 + 58.8i)T^{2}
61 1+(−1.96+9.17i)T+(−55.6−24.9i)T2 1 + (-1.96 + 9.17i)T + (-55.6 - 24.9i)T^{2}
67 1+(−8.97−8.36i)T+(4.70+66.8i)T2 1 + (-8.97 - 8.36i)T + (4.70 + 66.8i)T^{2}
71 1+(−0.00841+0.00308i)T+(54.1−45.9i)T2 1 + (-0.00841 + 0.00308i)T + (54.1 - 45.9i)T^{2}
73 1+(3.83+3.24i)T+(11.9+72.0i)T2 1 + (3.83 + 3.24i)T + (11.9 + 72.0i)T^{2}
79 1+(3.55+13.4i)T+(−68.7+38.9i)T2 1 + (3.55 + 13.4i)T + (-68.7 + 38.9i)T^{2}
83 1+(−0.183−0.967i)T+(−77.2+30.4i)T2 1 + (-0.183 - 0.967i)T + (-77.2 + 30.4i)T^{2}
89 1+(−4.94+1.68i)T+(70.5−54.3i)T2 1 + (-4.94 + 1.68i)T + (70.5 - 54.3i)T^{2}
97 1+(0.0510+0.238i)T+(−88.4+39.7i)T2 1 + (0.0510 + 0.238i)T + (-88.4 + 39.7i)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

−10.64970232685665432308864977163, −10.19376126193284128031836382913, −9.287081706692666180047954324414, −8.684821373650719004941002348303, −6.89323181971583717933539350307, −6.37451643326772851031884970150, −5.18436473242182650943996566583, −3.60835208693846960502878597698, −3.11960513005551406763910929441, −1.43220696990300565597797063796, 1.34851046341056459090630247040, 2.36582304449257446877960089416, 4.52763285299132586500099872778, 5.49419948598929492478695127177, 6.44487664467727120685020908853, 6.84493374848093035424475589005, 8.287577236283258976453453277826, 9.078595173574850684650504845550, 9.697729493243931114350545701745, 10.35440059920404706979022129753

Graph of the ZZ-function along the critical line