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Properties

Label 2-532-133.40-c1-0-12
Degree 22
Conductor 532532
Sign −0.969−0.244i-0.969 - 0.244i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.438 − 2.48i)3-s + (−1.03 − 0.181i)5-s + (−2.55 + 0.670i)7-s + (−3.18 − 1.15i)9-s − 2.06·11-s + (−0.478 + 0.401i)13-s + (−0.905 + 2.48i)15-s + (−1.43 − 3.93i)17-s + (−3.76 − 2.19i)19-s + (0.544 + 6.66i)21-s + (−4.26 + 3.57i)23-s + (−3.66 − 1.33i)25-s + (−0.490 + 0.849i)27-s + (7.92 − 1.39i)29-s + (−0.252 + 0.437i)31-s + ⋯
L(s)  = 1  + (0.253 − 1.43i)3-s + (−0.461 − 0.0813i)5-s + (−0.967 + 0.253i)7-s + (−1.06 − 0.386i)9-s − 0.622·11-s + (−0.132 + 0.111i)13-s + (−0.233 + 0.642i)15-s + (−0.347 − 0.955i)17-s + (−0.864 − 0.503i)19-s + (0.118 + 1.45i)21-s + (−0.888 + 0.745i)23-s + (−0.733 − 0.266i)25-s + (−0.0944 + 0.163i)27-s + (1.47 − 0.259i)29-s + (−0.0453 + 0.0786i)31-s + ⋯

Functional equation

Λ(s)=(532s/2 ΓC(s) L(s)=((−0.969−0.244i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(532s/2 ΓC(s+1/2) L(s)=((−0.969−0.244i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 532532    =    22⋅7⋅192^{2} \cdot 7 \cdot 19
Sign: −0.969−0.244i-0.969 - 0.244i
Analytic conductor: 4.248044.24804
Root analytic conductor: 2.061072.06107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ532(173,⋅)\chi_{532} (173, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 532, ( :1/2), −0.969−0.244i)(2,\ 532,\ (\ :1/2),\ -0.969 - 0.244i)

Particular Values

L(1)L(1) ≈\approx 0.0700646+0.565070i0.0700646 + 0.565070i
L(12)L(\frac12) ≈\approx 0.0700646+0.565070i0.0700646 + 0.565070i
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+(2.55−0.670i)T 1 + (2.55 - 0.670i)T
19 1+(3.76+2.19i)T 1 + (3.76 + 2.19i)T
good3 1+(−0.438+2.48i)T+(−2.81−1.02i)T2 1 + (-0.438 + 2.48i)T + (-2.81 - 1.02i)T^{2}
5 1+(1.03+0.181i)T+(4.69+1.71i)T2 1 + (1.03 + 0.181i)T + (4.69 + 1.71i)T^{2}
11 1+2.06T+11T2 1 + 2.06T + 11T^{2}
13 1+(0.478−0.401i)T+(2.25−12.8i)T2 1 + (0.478 - 0.401i)T + (2.25 - 12.8i)T^{2}
17 1+(1.43+3.93i)T+(−13.0+10.9i)T2 1 + (1.43 + 3.93i)T + (-13.0 + 10.9i)T^{2}
23 1+(4.26−3.57i)T+(3.99−22.6i)T2 1 + (4.26 - 3.57i)T + (3.99 - 22.6i)T^{2}
29 1+(−7.92+1.39i)T+(27.2−9.91i)T2 1 + (-7.92 + 1.39i)T + (27.2 - 9.91i)T^{2}
31 1+(0.252−0.437i)T+(−15.5−26.8i)T2 1 + (0.252 - 0.437i)T + (-15.5 - 26.8i)T^{2}
37 1+(−6.07−3.50i)T+(18.5+32.0i)T2 1 + (-6.07 - 3.50i)T + (18.5 + 32.0i)T^{2}
41 1+(−0.356−0.299i)T+(7.11+40.3i)T2 1 + (-0.356 - 0.299i)T + (7.11 + 40.3i)T^{2}
43 1+(8.38−3.05i)T+(32.9−27.6i)T2 1 + (8.38 - 3.05i)T + (32.9 - 27.6i)T^{2}
47 1+(−0.662+1.82i)T+(−36.0−30.2i)T2 1 + (-0.662 + 1.82i)T + (-36.0 - 30.2i)T^{2}
53 1+(0.276−0.0487i)T+(49.8−18.1i)T2 1 + (0.276 - 0.0487i)T + (49.8 - 18.1i)T^{2}
59 1+(−6.12+2.22i)T+(45.1−37.9i)T2 1 + (-6.12 + 2.22i)T + (45.1 - 37.9i)T^{2}
61 1+(7.44+8.87i)T+(−10.5+60.0i)T2 1 + (7.44 + 8.87i)T + (-10.5 + 60.0i)T^{2}
67 1+(3.76+4.49i)T+(−11.6+65.9i)T2 1 + (3.76 + 4.49i)T + (-11.6 + 65.9i)T^{2}
71 1+(3.35+9.22i)T+(−54.3+45.6i)T2 1 + (3.35 + 9.22i)T + (-54.3 + 45.6i)T^{2}
73 1+(−9.01−1.58i)T+(68.5+24.9i)T2 1 + (-9.01 - 1.58i)T + (68.5 + 24.9i)T^{2}
79 1+(4.23+11.6i)T+(−60.5+50.7i)T2 1 + (4.23 + 11.6i)T + (-60.5 + 50.7i)T^{2}
83 1+(−4.36+2.51i)T+(41.5−71.8i)T2 1 + (-4.36 + 2.51i)T + (41.5 - 71.8i)T^{2}
89 1+(2.04+11.6i)T+(−83.6+30.4i)T2 1 + (2.04 + 11.6i)T + (-83.6 + 30.4i)T^{2}
97 1+(−1.08+6.16i)T+(−91.1−33.1i)T2 1 + (-1.08 + 6.16i)T + (-91.1 - 33.1i)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

−10.31311277940106164443683036223, −9.375217016277851354285332202965, −8.304914520066700042915913001468, −7.68159231831007945965504554850, −6.72466482896591569889367499955, −6.12070638785765771309062178785, −4.65428260525579214164233216974, −3.11989778218925495553673982889, −2.14902185596829589340890210706, −0.30033857921307907045958076959, 2.66227270777722281946690513437, 3.83283540440384591990145126415, 4.34016776057669541120221196094, 5.67572784670606821691637998282, 6.67631404759489516036609637744, 8.020599178216914503912041050177, 8.721436772052140842508576744793, 9.829074792895999453101983193431, 10.29637850464245872477426500278, 10.91450135807758058240406264881

Graph of the ZZ-function along the critical line