↑

Properties

Label 2-532-133.10-c1-0-6
Degree 22
Conductor 532532
Sign 0.757+0.652i0.757 + 0.652i
Analytic cond. 4.248044.24804
Root an. cond. 2.061072.06107
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.317 − 1.79i)3-s + (2.76 − 0.487i)5-s + (1.96 + 1.76i)7-s + (−0.312 + 0.113i)9-s + 0.787·11-s + (3.33 + 2.79i)13-s + (−1.75 − 4.82i)15-s + (−1.10 + 3.04i)17-s + (−4.35 + 0.199i)19-s + (2.55 − 4.09i)21-s + (5.85 + 4.91i)23-s + (2.72 − 0.990i)25-s + (−2.43 − 4.21i)27-s + (−5.09 − 0.898i)29-s + (−5.06 − 8.76i)31-s + ⋯
L(s)  = 1  + (−0.183 − 1.03i)3-s + (1.23 − 0.218i)5-s + (0.743 + 0.668i)7-s + (−0.104 + 0.0379i)9-s + 0.237·11-s + (0.924 + 0.776i)13-s + (−0.453 − 1.24i)15-s + (−0.268 + 0.738i)17-s + (−0.998 + 0.0458i)19-s + (0.557 − 0.894i)21-s + (1.22 + 1.02i)23-s + (0.544 − 0.198i)25-s + (−0.468 − 0.811i)27-s + (−0.946 − 0.166i)29-s + (−0.908 − 1.57i)31-s + ⋯

Functional equation

Λ(s)=(532s/2 ΓC(s) L(s)=((0.757+0.652i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(532s/2 ΓC(s+1/2) L(s)=((0.757+0.652i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 532532    =    22⋅7⋅192^{2} \cdot 7 \cdot 19
Sign: 0.757+0.652i0.757 + 0.652i
Analytic conductor: 4.248044.24804
Root analytic conductor: 2.061072.06107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ532(409,⋅)\chi_{532} (409, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 532, ( :1/2), 0.757+0.652i)(2,\ 532,\ (\ :1/2),\ 0.757 + 0.652i)

Particular Values

L(1)L(1) ≈\approx 1.75020−0.649943i1.75020 - 0.649943i
L(12)L(\frac12) ≈\approx 1.75020−0.649943i1.75020 - 0.649943i
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 1
7 1+(−1.96−1.76i)T 1 + (-1.96 - 1.76i)T
19 1+(4.35−0.199i)T 1 + (4.35 - 0.199i)T
good3 1+(0.317+1.79i)T+(−2.81+1.02i)T2 1 + (0.317 + 1.79i)T + (-2.81 + 1.02i)T^{2}
5 1+(−2.76+0.487i)T+(4.69−1.71i)T2 1 + (-2.76 + 0.487i)T + (4.69 - 1.71i)T^{2}
11 1−0.787T+11T2 1 - 0.787T + 11T^{2}
13 1+(−3.33−2.79i)T+(2.25+12.8i)T2 1 + (-3.33 - 2.79i)T + (2.25 + 12.8i)T^{2}
17 1+(1.10−3.04i)T+(−13.0−10.9i)T2 1 + (1.10 - 3.04i)T + (-13.0 - 10.9i)T^{2}
23 1+(−5.85−4.91i)T+(3.99+22.6i)T2 1 + (-5.85 - 4.91i)T + (3.99 + 22.6i)T^{2}
29 1+(5.09+0.898i)T+(27.2+9.91i)T2 1 + (5.09 + 0.898i)T + (27.2 + 9.91i)T^{2}
31 1+(5.06+8.76i)T+(−15.5+26.8i)T2 1 + (5.06 + 8.76i)T + (-15.5 + 26.8i)T^{2}
37 1+(−4.14+2.39i)T+(18.5−32.0i)T2 1 + (-4.14 + 2.39i)T + (18.5 - 32.0i)T^{2}
41 1+(−2.63+2.21i)T+(7.11−40.3i)T2 1 + (-2.63 + 2.21i)T + (7.11 - 40.3i)T^{2}
43 1+(9.94+3.61i)T+(32.9+27.6i)T2 1 + (9.94 + 3.61i)T + (32.9 + 27.6i)T^{2}
47 1+(1.22+3.37i)T+(−36.0+30.2i)T2 1 + (1.22 + 3.37i)T + (-36.0 + 30.2i)T^{2}
53 1+(1.53+0.270i)T+(49.8+18.1i)T2 1 + (1.53 + 0.270i)T + (49.8 + 18.1i)T^{2}
59 1+(8.34+3.03i)T+(45.1+37.9i)T2 1 + (8.34 + 3.03i)T + (45.1 + 37.9i)T^{2}
61 1+(−5.16+6.16i)T+(−10.5−60.0i)T2 1 + (-5.16 + 6.16i)T + (-10.5 - 60.0i)T^{2}
67 1+(1.41−1.68i)T+(−11.6−65.9i)T2 1 + (1.41 - 1.68i)T + (-11.6 - 65.9i)T^{2}
71 1+(1.65−4.53i)T+(−54.3−45.6i)T2 1 + (1.65 - 4.53i)T + (-54.3 - 45.6i)T^{2}
73 1+(−1.88+0.332i)T+(68.5−24.9i)T2 1 + (-1.88 + 0.332i)T + (68.5 - 24.9i)T^{2}
79 1+(4.67−12.8i)T+(−60.5−50.7i)T2 1 + (4.67 - 12.8i)T + (-60.5 - 50.7i)T^{2}
83 1+(10.3+5.95i)T+(41.5+71.8i)T2 1 + (10.3 + 5.95i)T + (41.5 + 71.8i)T^{2}
89 1+(2.43−13.7i)T+(−83.6−30.4i)T2 1 + (2.43 - 13.7i)T + (-83.6 - 30.4i)T^{2}
97 1+(−0.732−4.15i)T+(−91.1+33.1i)T2 1 + (-0.732 - 4.15i)T + (-91.1 + 33.1i)T^{2}
show more
show less
   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

−11.03326734451756459753232807516, −9.643458672065517249716765908901, −8.987127836731877913481372515856, −8.085314729257364477453175956209, −6.95146244160962615000827990707, −6.10107850914392689410194012127, −5.49936464160248222723824297302, −4.05384203982370486033786721443, −2.05481628680848780263520441987, −1.58353266323813960664603986168, 1.54506033713819144577373520171, 3.15676249078125557146096023230, 4.45261931934496136301975884146, 5.16609695802949302334456938075, 6.22318585910730286915191184872, 7.21918210193565470085658648903, 8.551433221435037472924400780547, 9.300566967942060095065358443632, 10.27854031289025364069814545798, 10.71205138937984194891926500349

Graph of the ZZ-function along the critical line