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Properties

Label 2-539-7.2-c1-0-5
Degree 22
Conductor 539539
Sign −0.968+0.250i-0.968 + 0.250i
Analytic cond. 4.303934.30393
Root an. cond. 2.074592.07459
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  + (−0.5 + 0.866i)2-s + (1 + 1.73i)3-s + (0.500 + 0.866i)4-s + (−1 + 1.73i)5-s − 1.99·6-s − 3·8-s + (−0.499 + 0.866i)9-s + (−0.999 − 1.73i)10-s + (−0.5 − 0.866i)11-s + (−0.999 + 1.73i)12-s − 4·13-s − 3.99·15-s + (0.500 − 0.866i)16-s + (2 + 3.46i)17-s + (−0.499 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.577 + 0.999i)3-s + (0.250 + 0.433i)4-s + (−0.447 + 0.774i)5-s − 0.816·6-s − 1.06·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (−0.150 − 0.261i)11-s + (−0.288 + 0.499i)12-s − 1.10·13-s − 1.03·15-s + (0.125 − 0.216i)16-s + (0.485 + 0.840i)17-s + (−0.117 − 0.204i)18-s + ⋯

Functional equation

Λ(s)=(539s/2 ΓC(s) L(s)=((−0.968+0.250i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(539s/2 ΓC(s+1/2) L(s)=((−0.968+0.250i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 539539    =    72⋅117^{2} \cdot 11
Sign: −0.968+0.250i-0.968 + 0.250i
Analytic conductor: 4.303934.30393
Root analytic conductor: 2.074592.07459
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ539(177,⋅)\chi_{539} (177, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 539, ( :1/2), −0.968+0.250i)(2,\ 539,\ (\ :1/2),\ -0.968 + 0.250i)

Particular Values

L(1)L(1) ≈\approx 0.148273−1.16353i0.148273 - 1.16353i
L(12)L(\frac12) ≈\approx 0.148273−1.16353i0.148273 - 1.16353i
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)
bad7 1 1
11 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(0.5−0.866i)T+(−1−1.73i)T2 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2}
3 1+(−1−1.73i)T+(−1.5+2.59i)T2 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2}
5 1+(1−1.73i)T+(−2.5−4.33i)T2 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2}
13 1+4T+13T2 1 + 4T + 13T^{2}
17 1+(−2−3.46i)T+(−8.5+14.7i)T2 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2}
19 1+(−9.5−16.4i)T2 1 + (-9.5 - 16.4i)T^{2}
23 1+(−2+3.46i)T+(−11.5−19.9i)T2 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1+(−5−8.66i)T+(−15.5+26.8i)T2 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2}
37 1+(−3+5.19i)T+(−18.5−32.0i)T2 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2}
41 1+4T+41T2 1 + 4T + 41T^{2}
43 1−12T+43T2 1 - 12T + 43T^{2}
47 1+(5−8.66i)T+(−23.5−40.7i)T2 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2}
53 1+(−3−5.19i)T+(−26.5+45.8i)T2 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1−1.73i)T+(−29.5+51.0i)T2 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2}
61 1+(−30.5−52.8i)T2 1 + (-30.5 - 52.8i)T^{2}
67 1+(4+6.92i)T+(−33.5+58.0i)T2 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 1+(4+6.92i)T+(−36.5+63.2i)T2 1 + (4 + 6.92i)T + (-36.5 + 63.2i)T^{2}
79 1+(4−6.92i)T+(−39.5−68.4i)T2 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2}
83 1+83T2 1 + 83T^{2}
89 1+(3−5.19i)T+(−44.5−77.0i)T2 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2}
97 1−10T+97T2 1 - 10T + 97T^{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

−11.01399970901142558658085411459, −10.35417603145711302199356990572, −9.360059878873926623215291788329, −8.654573274737109421972609295479, −7.69088727673016874897390417370, −7.05166387642353150669127418995, −5.95800371243950598455606320016, −4.53986068225512843222698464083, −3.43296212852972034518347041345, −2.75367979119873177476454494544, 0.69879789330897275188748878624, 1.98172644318346283194013118055, 2.92424844745527551945513742561, 4.62513542852633258320781217837, 5.69366575655268010007718462092, 7.04171009416625584697534537626, 7.65390531723111661333422869106, 8.604315995159997365852291002654, 9.531001684614036680414375611918, 10.16352196279216803423384725093

Graph of the ZZ-function along the critical line