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Properties

Label 2-532-133.10-c1-0-0
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(409,⋅)\chi_{532} (409, \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

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

Graph of the ZZ-function along the critical line