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Properties

Label 2-538-269.191-c1-0-7
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(191,⋅)\chi_{538} (191, \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

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

Graph of the ZZ-function along the critical line