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Properties

Label 2-538-269.191-c1-0-8
Degree 22
Conductor 538538
Sign 0.937−0.348i0.937 - 0.348i
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 + (−2.31 + 0.0543i)3-s + (−0.930 + 0.366i)4-s + (−0.224 + 0.850i)5-s + (−0.485 − 2.26i)6-s + (−1.08 − 2.42i)7-s + (−0.533 − 0.845i)8-s + (2.36 − 0.111i)9-s + (−0.877 − 0.0618i)10-s + (2.35 + 2.64i)11-s + (2.13 − 0.899i)12-s + (0.370 − 5.25i)13-s + (2.18 − 1.52i)14-s + (0.473 − 1.98i)15-s + (0.731 − 0.681i)16-s + (−3.40 − 3.02i)17-s + ⋯
L(s)  = 1  + (0.131 + 0.694i)2-s + (−1.33 + 0.0313i)3-s + (−0.465 + 0.183i)4-s + (−0.100 + 0.380i)5-s + (−0.198 − 0.925i)6-s + (−0.411 − 0.917i)7-s + (−0.188 − 0.299i)8-s + (0.789 − 0.0370i)9-s + (−0.277 − 0.0195i)10-s + (0.709 + 0.797i)11-s + (0.616 − 0.259i)12-s + (0.102 − 1.45i)13-s + (0.582 − 0.407i)14-s + (0.122 − 0.511i)15-s + (0.182 − 0.170i)16-s + (−0.825 − 0.734i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((0.937−0.348i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((0.937−0.348i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.937−0.348i0.937 - 0.348i
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), 0.937−0.348i)(2,\ 538,\ (\ :1/2),\ 0.937 - 0.348i)

Particular Values

L(1)L(1) ≈\approx 0.809343+0.145539i0.809343 + 0.145539i
L(12)L(\frac12) ≈\approx 0.809343+0.145539i0.809343 + 0.145539i
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+(15.4−5.47i)T 1 + (15.4 - 5.47i)T
good3 1+(2.31−0.0543i)T+(2.99−0.140i)T2 1 + (2.31 - 0.0543i)T + (2.99 - 0.140i)T^{2}
5 1+(0.224−0.850i)T+(−4.34−2.46i)T2 1 + (0.224 - 0.850i)T + (-4.34 - 2.46i)T^{2}
7 1+(1.08+2.42i)T+(−4.65+5.23i)T2 1 + (1.08 + 2.42i)T + (-4.65 + 5.23i)T^{2}
11 1+(−2.35−2.64i)T+(−1.28+10.9i)T2 1 + (-2.35 - 2.64i)T + (-1.28 + 10.9i)T^{2}
13 1+(−0.370+5.25i)T+(−12.8−1.82i)T2 1 + (-0.370 + 5.25i)T + (-12.8 - 1.82i)T^{2}
17 1+(3.40+3.02i)T+(1.98+16.8i)T2 1 + (3.40 + 3.02i)T + (1.98 + 16.8i)T^{2}
19 1+(−4.24−2.81i)T+(7.37+17.5i)T2 1 + (-4.24 - 2.81i)T + (7.37 + 17.5i)T^{2}
23 1+(−0.134−5.73i)T+(−22.9+1.07i)T2 1 + (-0.134 - 5.73i)T + (-22.9 + 1.07i)T^{2}
29 1+(0.938+1.65i)T+(−14.8+24.8i)T2 1 + (0.938 + 1.65i)T + (-14.8 + 24.8i)T^{2}
31 1+(−9.45−1.11i)T+(30.1+7.20i)T2 1 + (-9.45 - 1.11i)T + (30.1 + 7.20i)T^{2}
37 1+(−4.91+9.15i)T+(−20.4−30.8i)T2 1 + (-4.91 + 9.15i)T + (-20.4 - 30.8i)T^{2}
41 1+(3.18+0.604i)T+(38.1+15.0i)T2 1 + (3.18 + 0.604i)T + (38.1 + 15.0i)T^{2}
43 1+(−3.84+2.18i)T+(22.0−36.8i)T2 1 + (-3.84 + 2.18i)T + (22.0 - 36.8i)T^{2}
47 1+(3.58+1.03i)T+(39.7+25.0i)T2 1 + (3.58 + 1.03i)T + (39.7 + 25.0i)T^{2}
53 1+(−5.51−7.51i)T+(−15.9+50.5i)T2 1 + (-5.51 - 7.51i)T + (-15.9 + 50.5i)T^{2}
59 1+(−0.291−0.741i)T+(−43.1+40.2i)T2 1 + (-0.291 - 0.741i)T + (-43.1 + 40.2i)T^{2}
61 1+(5.14+6.36i)T+(−12.7+59.6i)T2 1 + (5.14 + 6.36i)T + (-12.7 + 59.6i)T^{2}
67 1+(9.93+3.91i)T+(49.0+45.6i)T2 1 + (9.93 + 3.91i)T + (49.0 + 45.6i)T^{2}
71 1+(−6.15+8.80i)T+(−24.4−66.6i)T2 1 + (-6.15 + 8.80i)T + (-24.4 - 66.6i)T^{2}
73 1+(−1.69+4.62i)T+(−55.6−47.2i)T2 1 + (-1.69 + 4.62i)T + (-55.6 - 47.2i)T^{2}
79 1+(−10.5−8.12i)T+(20.1+76.3i)T2 1 + (-10.5 - 8.12i)T + (20.1 + 76.3i)T^{2}
83 1+(0.492+5.24i)T+(−81.5+15.4i)T2 1 + (0.492 + 5.24i)T + (-81.5 + 15.4i)T^{2}
89 1+(0.360+2.17i)T+(−84.2+28.6i)T2 1 + (0.360 + 2.17i)T + (-84.2 + 28.6i)T^{2}
97 1+(3.09−3.82i)T+(−20.3−94.8i)T2 1 + (3.09 - 3.82i)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

−10.83340818191169550747075380964, −10.12540639165366927930994002441, −9.268560249097454237593635163216, −7.71590368390540616524944364146, −7.13020118240413045475435688325, −6.29600695688783146403870646369, −5.43591584417629777389896863747, −4.50144446621961414284731396104, −3.31456990063129898345879765095, −0.74501494333988854069203815394, 1.02109374203441172486044433140, 2.71286033652376579619764961399, 4.28586965716443235187510905871, 5.02157907412717997624765965637, 6.31824728665021883231346273349, 6.49782390309122493345170206128, 8.540242371888770456365351151518, 9.027678717277635103882505020338, 10.08753942311236449225746424034, 11.11631415700135932182268949323

Graph of the ZZ-function along the critical line