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Properties

Label 2-532-133.102-c1-0-7
Degree 22
Conductor 532532
Sign 0.533+0.846i0.533 + 0.846i
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  − 1.07·3-s + (−1.76 + 3.05i)5-s + (−1.25 − 2.32i)7-s − 1.84·9-s + (2.10 − 3.64i)11-s + (0.480 − 0.831i)13-s + (1.89 − 3.27i)15-s + 4.91·17-s + (2.82 − 3.32i)19-s + (1.34 + 2.49i)21-s + 3.66·23-s + (−3.70 − 6.41i)25-s + 5.20·27-s + (2.27 − 3.93i)29-s + (−3.20 + 5.54i)31-s + ⋯
L(s)  = 1  − 0.619·3-s + (−0.787 + 1.36i)5-s + (−0.475 − 0.879i)7-s − 0.615·9-s + (0.635 − 1.09i)11-s + (0.133 − 0.230i)13-s + (0.488 − 0.845i)15-s + 1.19·17-s + (0.647 − 0.762i)19-s + (0.294 + 0.545i)21-s + 0.765·23-s + (−0.740 − 1.28i)25-s + 1.00·27-s + (0.421 − 0.730i)29-s + (−0.574 + 0.995i)31-s + ⋯

Functional equation

Λ(s)=(532s/2 ΓC(s) L(s)=((0.533+0.846i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(532s/2 ΓC(s+1/2) L(s)=((0.533+0.846i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 532532    =    22⋅7⋅192^{2} \cdot 7 \cdot 19
Sign: 0.533+0.846i0.533 + 0.846i
Analytic conductor: 4.248044.24804
Root analytic conductor: 2.061072.06107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ532(501,⋅)\chi_{532} (501, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 532, ( :1/2), 0.533+0.846i)(2,\ 532,\ (\ :1/2),\ 0.533 + 0.846i)

Particular Values

L(1)L(1) ≈\approx 0.673863−0.371888i0.673863 - 0.371888i
L(12)L(\frac12) ≈\approx 0.673863−0.371888i0.673863 - 0.371888i
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+(1.25+2.32i)T 1 + (1.25 + 2.32i)T
19 1+(−2.82+3.32i)T 1 + (-2.82 + 3.32i)T
good3 1+1.07T+3T2 1 + 1.07T + 3T^{2}
5 1+(1.76−3.05i)T+(−2.5−4.33i)T2 1 + (1.76 - 3.05i)T + (-2.5 - 4.33i)T^{2}
11 1+(−2.10+3.64i)T+(−5.5−9.52i)T2 1 + (-2.10 + 3.64i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.480+0.831i)T+(−6.5−11.2i)T2 1 + (-0.480 + 0.831i)T + (-6.5 - 11.2i)T^{2}
17 1−4.91T+17T2 1 - 4.91T + 17T^{2}
23 1−3.66T+23T2 1 - 3.66T + 23T^{2}
29 1+(−2.27+3.93i)T+(−14.5−25.1i)T2 1 + (-2.27 + 3.93i)T + (-14.5 - 25.1i)T^{2}
31 1+(3.20−5.54i)T+(−15.5−26.8i)T2 1 + (3.20 - 5.54i)T + (-15.5 - 26.8i)T^{2}
37 1+(−1.28−2.21i)T+(−18.5+32.0i)T2 1 + (-1.28 - 2.21i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.84+6.66i)T+(−20.5+35.5i)T2 1 + (3.84 + 6.66i)T + (-20.5 + 35.5i)T^{2}
43 1+(5.27+9.13i)T+(−21.5+37.2i)T2 1 + (5.27 + 9.13i)T + (-21.5 + 37.2i)T^{2}
47 1+0.902T+47T2 1 + 0.902T + 47T^{2}
53 1+(6.91+11.9i)T+(−26.5+45.8i)T2 1 + (6.91 + 11.9i)T + (-26.5 + 45.8i)T^{2}
59 1−7.62T+59T2 1 - 7.62T + 59T^{2}
61 1+2.34T+61T2 1 + 2.34T + 61T^{2}
67 1+(5.49+9.51i)T+(−33.5+58.0i)T2 1 + (5.49 + 9.51i)T + (-33.5 + 58.0i)T^{2}
71 1+(−0.929−1.61i)T+(−35.5+61.4i)T2 1 + (-0.929 - 1.61i)T + (-35.5 + 61.4i)T^{2}
73 1+9.95T+73T2 1 + 9.95T + 73T^{2}
79 1+(−4.45+7.71i)T+(−39.5−68.4i)T2 1 + (-4.45 + 7.71i)T + (-39.5 - 68.4i)T^{2}
83 1+0.481T+83T2 1 + 0.481T + 83T^{2}
89 1−17.2T+89T2 1 - 17.2T + 89T^{2}
97 1+(−2.65−4.59i)T+(−48.5+84.0i)T2 1 + (-2.65 - 4.59i)T + (-48.5 + 84.0i)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

−10.74899170103110915431430416683, −10.22342875576444697011411698855, −8.892985510949621311332412250415, −7.80764979844375728627142453932, −6.92682554221899713715117194029, −6.31709310508187046483550590096, −5.17341514388119166489894926270, −3.48583945544305154810449393419, −3.21913851400453235787585936104, −0.55888258520169276543178008862, 1.29155937399692528818197409037, 3.23201370463740731582455023147, 4.54032141658779930620317973072, 5.33442570144776787139411567738, 6.18886753964036697564925892593, 7.47786922956026231377651401704, 8.406320296087745075112892353650, 9.204557575829916443358268058046, 9.895480948427269331362191741892, 11.36020086815316711243472057046

Graph of the ZZ-function along the critical line