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Properties

Label 2-538-269.100-c1-0-17
Degree 22
Conductor 538538
Sign 0.973+0.229i0.973 + 0.229i
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  + (−0.186 + 0.982i)2-s + (0.932 + 0.0218i)3-s + (−0.930 − 0.366i)4-s + (0.272 + 1.03i)5-s + (−0.195 + 0.912i)6-s + (1.94 − 4.33i)7-s + (0.533 − 0.845i)8-s + (−2.12 − 0.0998i)9-s + (−1.06 + 0.0752i)10-s + (3.06 − 3.44i)11-s + (−0.860 − 0.362i)12-s + (−0.431 − 6.12i)13-s + (3.89 + 2.72i)14-s + (0.232 + 0.971i)15-s + (0.731 + 0.681i)16-s + (−3.31 + 2.94i)17-s + ⋯
L(s)  = 1  + (−0.131 + 0.694i)2-s + (0.538 + 0.0126i)3-s + (−0.465 − 0.183i)4-s + (0.122 + 0.462i)5-s + (−0.0797 + 0.372i)6-s + (0.735 − 1.63i)7-s + (0.188 − 0.299i)8-s + (−0.708 − 0.0332i)9-s + (−0.337 + 0.0237i)10-s + (0.922 − 1.03i)11-s + (−0.248 − 0.104i)12-s + (−0.119 − 1.69i)13-s + (1.04 + 0.727i)14-s + (0.0599 + 0.250i)15-s + (0.182 + 0.170i)16-s + (−0.802 + 0.713i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((0.973+0.229i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((0.973+0.229i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.973+0.229i0.973 + 0.229i
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ538(369,⋅)\chi_{538} (369, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), 0.973+0.229i)(2,\ 538,\ (\ :1/2),\ 0.973 + 0.229i)

Particular Values

L(1)L(1) ≈\approx 1.55797−0.181133i1.55797 - 0.181133i
L(12)L(\frac12) ≈\approx 1.55797−0.181133i1.55797 - 0.181133i
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+(0.186−0.982i)T 1 + (0.186 - 0.982i)T
269 1+(−16.0−3.51i)T 1 + (-16.0 - 3.51i)T
good3 1+(−0.932−0.0218i)T+(2.99+0.140i)T2 1 + (-0.932 - 0.0218i)T + (2.99 + 0.140i)T^{2}
5 1+(−0.272−1.03i)T+(−4.34+2.46i)T2 1 + (-0.272 - 1.03i)T + (-4.34 + 2.46i)T^{2}
7 1+(−1.94+4.33i)T+(−4.65−5.23i)T2 1 + (-1.94 + 4.33i)T + (-4.65 - 5.23i)T^{2}
11 1+(−3.06+3.44i)T+(−1.28−10.9i)T2 1 + (-3.06 + 3.44i)T + (-1.28 - 10.9i)T^{2}
13 1+(0.431+6.12i)T+(−12.8+1.82i)T2 1 + (0.431 + 6.12i)T + (-12.8 + 1.82i)T^{2}
17 1+(3.31−2.94i)T+(1.98−16.8i)T2 1 + (3.31 - 2.94i)T + (1.98 - 16.8i)T^{2}
19 1+(−1.25+0.831i)T+(7.37−17.5i)T2 1 + (-1.25 + 0.831i)T + (7.37 - 17.5i)T^{2}
23 1+(0.0343−1.46i)T+(−22.9−1.07i)T2 1 + (0.0343 - 1.46i)T + (-22.9 - 1.07i)T^{2}
29 1+(4.24−7.48i)T+(−14.8−24.8i)T2 1 + (4.24 - 7.48i)T + (-14.8 - 24.8i)T^{2}
31 1+(−3.16+0.372i)T+(30.1−7.20i)T2 1 + (-3.16 + 0.372i)T + (30.1 - 7.20i)T^{2}
37 1+(−3.72−6.95i)T+(−20.4+30.8i)T2 1 + (-3.72 - 6.95i)T + (-20.4 + 30.8i)T^{2}
41 1+(1.94−0.368i)T+(38.1−15.0i)T2 1 + (1.94 - 0.368i)T + (38.1 - 15.0i)T^{2}
43 1+(−2.13−1.21i)T+(22.0+36.8i)T2 1 + (-2.13 - 1.21i)T + (22.0 + 36.8i)T^{2}
47 1+(−9.96+2.87i)T+(39.7−25.0i)T2 1 + (-9.96 + 2.87i)T + (39.7 - 25.0i)T^{2}
53 1+(5.13−7.00i)T+(−15.9−50.5i)T2 1 + (5.13 - 7.00i)T + (-15.9 - 50.5i)T^{2}
59 1+(−1.32+3.36i)T+(−43.1−40.2i)T2 1 + (-1.32 + 3.36i)T + (-43.1 - 40.2i)T^{2}
61 1+(−5.85+7.23i)T+(−12.7−59.6i)T2 1 + (-5.85 + 7.23i)T + (-12.7 - 59.6i)T^{2}
67 1+(2.45−0.966i)T+(49.0−45.6i)T2 1 + (2.45 - 0.966i)T + (49.0 - 45.6i)T^{2}
71 1+(−3.25−4.66i)T+(−24.4+66.6i)T2 1 + (-3.25 - 4.66i)T + (-24.4 + 66.6i)T^{2}
73 1+(−3.48−9.49i)T+(−55.6+47.2i)T2 1 + (-3.48 - 9.49i)T + (-55.6 + 47.2i)T^{2}
79 1+(2.89−2.22i)T+(20.1−76.3i)T2 1 + (2.89 - 2.22i)T + (20.1 - 76.3i)T^{2}
83 1+(1.59−16.9i)T+(−81.5−15.4i)T2 1 + (1.59 - 16.9i)T + (-81.5 - 15.4i)T^{2}
89 1+(−1.09+6.60i)T+(−84.2−28.6i)T2 1 + (-1.09 + 6.60i)T + (-84.2 - 28.6i)T^{2}
97 1+(−0.866−1.07i)T+(−20.3+94.8i)T2 1 + (-0.866 - 1.07i)T + (-20.3 + 94.8i)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

−10.84885147835766412937316808385, −9.896014843895020532362659384752, −8.662602886100140163300788377982, −8.154696757542749368890199779753, −7.27529428921014756817024427639, −6.36668631939220063324033012079, −5.30207440443958534585414153450, −3.99842176828936225943741763163, −3.09438439911888905429143912854, −0.969064439609761032719147990020, 1.91111587342730472084300820732, 2.45434203971192553439981055752, 4.15742498510686828762704133521, 4.98951943434619601423406850163, 6.15547444704976669579265884475, 7.49130305566079557105451755453, 8.702844788240429849466397082896, 9.116663730968845359093888466719, 9.513581416904266318058227832198, 11.20115283761584145284936982606

Graph of the ZZ-function along the critical line