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Properties

Label 2-538-269.191-c1-0-18
Degree 22
Conductor 538538
Sign −0.904+0.427i-0.904 + 0.427i
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 + (0.411 − 0.00964i)3-s + (−0.930 + 0.366i)4-s + (0.415 − 1.57i)5-s + (−0.0862 − 0.402i)6-s + (−0.208 − 0.464i)7-s + (0.533 + 0.845i)8-s + (−2.82 + 0.132i)9-s + (−1.62 − 0.114i)10-s + (−0.181 − 0.203i)11-s + (−0.379 + 0.159i)12-s + (0.295 − 4.19i)13-s + (−0.417 + 0.291i)14-s + (0.155 − 0.652i)15-s + (0.731 − 0.681i)16-s + (−2.44 − 2.17i)17-s + ⋯
L(s)  = 1  + (−0.131 − 0.694i)2-s + (0.237 − 0.00557i)3-s + (−0.465 + 0.183i)4-s + (0.185 − 0.704i)5-s + (−0.0351 − 0.164i)6-s + (−0.0787 − 0.175i)7-s + (0.188 + 0.299i)8-s + (−0.942 + 0.0442i)9-s + (−0.514 − 0.0362i)10-s + (−0.0546 − 0.0614i)11-s + (−0.109 + 0.0461i)12-s + (0.0819 − 1.16i)13-s + (−0.111 + 0.0778i)14-s + (0.0402 − 0.168i)15-s + (0.182 − 0.170i)16-s + (−0.593 − 0.527i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((−0.904+0.427i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((−0.904+0.427i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −0.904+0.427i-0.904 + 0.427i
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), −0.904+0.427i)(2,\ 538,\ (\ :1/2),\ -0.904 + 0.427i)

Particular Values

L(1)L(1) ≈\approx 0.217824−0.971037i0.217824 - 0.971037i
L(12)L(\frac12) ≈\approx 0.217824−0.971037i0.217824 - 0.971037i
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+(14.9−6.77i)T 1 + (14.9 - 6.77i)T
good3 1+(−0.411+0.00964i)T+(2.99−0.140i)T2 1 + (-0.411 + 0.00964i)T + (2.99 - 0.140i)T^{2}
5 1+(−0.415+1.57i)T+(−4.34−2.46i)T2 1 + (-0.415 + 1.57i)T + (-4.34 - 2.46i)T^{2}
7 1+(0.208+0.464i)T+(−4.65+5.23i)T2 1 + (0.208 + 0.464i)T + (-4.65 + 5.23i)T^{2}
11 1+(0.181+0.203i)T+(−1.28+10.9i)T2 1 + (0.181 + 0.203i)T + (-1.28 + 10.9i)T^{2}
13 1+(−0.295+4.19i)T+(−12.8−1.82i)T2 1 + (-0.295 + 4.19i)T + (-12.8 - 1.82i)T^{2}
17 1+(2.44+2.17i)T+(1.98+16.8i)T2 1 + (2.44 + 2.17i)T + (1.98 + 16.8i)T^{2}
19 1+(2.59+1.72i)T+(7.37+17.5i)T2 1 + (2.59 + 1.72i)T + (7.37 + 17.5i)T^{2}
23 1+(0.141+6.02i)T+(−22.9+1.07i)T2 1 + (0.141 + 6.02i)T + (-22.9 + 1.07i)T^{2}
29 1+(−3.87−6.83i)T+(−14.8+24.8i)T2 1 + (-3.87 - 6.83i)T + (-14.8 + 24.8i)T^{2}
31 1+(4.64+0.547i)T+(30.1+7.20i)T2 1 + (4.64 + 0.547i)T + (30.1 + 7.20i)T^{2}
37 1+(−3.47+6.47i)T+(−20.4−30.8i)T2 1 + (-3.47 + 6.47i)T + (-20.4 - 30.8i)T^{2}
41 1+(2.28+0.434i)T+(38.1+15.0i)T2 1 + (2.28 + 0.434i)T + (38.1 + 15.0i)T^{2}
43 1+(3.06−1.74i)T+(22.0−36.8i)T2 1 + (3.06 - 1.74i)T + (22.0 - 36.8i)T^{2}
47 1+(3.54+1.02i)T+(39.7+25.0i)T2 1 + (3.54 + 1.02i)T + (39.7 + 25.0i)T^{2}
53 1+(2.96+4.04i)T+(−15.9+50.5i)T2 1 + (2.96 + 4.04i)T + (-15.9 + 50.5i)T^{2}
59 1+(−4.59−11.6i)T+(−43.1+40.2i)T2 1 + (-4.59 - 11.6i)T + (-43.1 + 40.2i)T^{2}
61 1+(6.85+8.47i)T+(−12.7+59.6i)T2 1 + (6.85 + 8.47i)T + (-12.7 + 59.6i)T^{2}
67 1+(−6.35−2.50i)T+(49.0+45.6i)T2 1 + (-6.35 - 2.50i)T + (49.0 + 45.6i)T^{2}
71 1+(−3.21+4.60i)T+(−24.4−66.6i)T2 1 + (-3.21 + 4.60i)T + (-24.4 - 66.6i)T^{2}
73 1+(−0.0380+0.103i)T+(−55.6−47.2i)T2 1 + (-0.0380 + 0.103i)T + (-55.6 - 47.2i)T^{2}
79 1+(−11.4−8.83i)T+(20.1+76.3i)T2 1 + (-11.4 - 8.83i)T + (20.1 + 76.3i)T^{2}
83 1+(−0.934−9.94i)T+(−81.5+15.4i)T2 1 + (-0.934 - 9.94i)T + (-81.5 + 15.4i)T^{2}
89 1+(−2.53−15.2i)T+(−84.2+28.6i)T2 1 + (-2.53 - 15.2i)T + (-84.2 + 28.6i)T^{2}
97 1+(−6.09+7.53i)T+(−20.3−94.8i)T2 1 + (-6.09 + 7.53i)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

−10.65442932319714107479884890314, −9.487355071993917201366428699410, −8.698422045191316289539922249008, −8.207039111721460865732260308444, −6.84763157147800633758048687282, −5.52916907296430524483392026978, −4.73007521657750157799894401335, −3.36261983895167198832315924409, −2.34315022262253750290728494954, −0.56903264891795374606393689674, 2.11637069879361549567020338768, 3.47140303092021573753283060279, 4.69519590167467681597091296977, 6.04718611496171052842888726353, 6.49459253890644687373749002685, 7.62602485829648389048484438256, 8.536520985806146910612140746538, 9.255779209490577090170533746453, 10.18326365222849336576306552073, 11.17586498200469795656764898161

Graph of the ZZ-function along the critical line