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Properties

Label 2-532-133.10-c1-0-10
Degree 22
Conductor 532532
Sign 0.614+0.789i0.614 + 0.789i
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  + (−0.0621 − 0.352i)3-s + (2.12 − 0.375i)5-s + (0.943 − 2.47i)7-s + (2.69 − 0.982i)9-s − 3.77·11-s + (0.242 + 0.203i)13-s + (−0.264 − 0.726i)15-s + (−0.714 + 1.96i)17-s + (2.16 − 3.78i)19-s + (−0.929 − 0.179i)21-s + (−2.05 − 1.72i)23-s + (−0.315 + 0.114i)25-s + (−1.05 − 1.81i)27-s + (3.99 + 0.703i)29-s + (1.54 + 2.67i)31-s + ⋯
L(s)  = 1  + (−0.0358 − 0.203i)3-s + (0.951 − 0.167i)5-s + (0.356 − 0.934i)7-s + (0.899 − 0.327i)9-s − 1.13·11-s + (0.0672 + 0.0564i)13-s + (−0.0682 − 0.187i)15-s + (−0.173 + 0.476i)17-s + (0.497 − 0.867i)19-s + (−0.202 − 0.0390i)21-s + (−0.429 − 0.360i)23-s + (−0.0631 + 0.0229i)25-s + (−0.202 − 0.350i)27-s + (0.741 + 0.130i)29-s + (0.276 + 0.479i)31-s + ⋯

Functional equation

Λ(s)=(532s/2 ΓC(s) L(s)=((0.614+0.789i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(532s/2 ΓC(s+1/2) L(s)=((0.614+0.789i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 532532    =    22⋅7⋅192^{2} \cdot 7 \cdot 19
Sign: 0.614+0.789i0.614 + 0.789i
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), 0.614+0.789i)(2,\ 532,\ (\ :1/2),\ 0.614 + 0.789i)

Particular Values

L(1)L(1) ≈\approx 1.54225−0.754209i1.54225 - 0.754209i
L(12)L(\frac12) ≈\approx 1.54225−0.754209i1.54225 - 0.754209i
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+(−0.943+2.47i)T 1 + (-0.943 + 2.47i)T
19 1+(−2.16+3.78i)T 1 + (-2.16 + 3.78i)T
good3 1+(0.0621+0.352i)T+(−2.81+1.02i)T2 1 + (0.0621 + 0.352i)T + (-2.81 + 1.02i)T^{2}
5 1+(−2.12+0.375i)T+(4.69−1.71i)T2 1 + (-2.12 + 0.375i)T + (4.69 - 1.71i)T^{2}
11 1+3.77T+11T2 1 + 3.77T + 11T^{2}
13 1+(−0.242−0.203i)T+(2.25+12.8i)T2 1 + (-0.242 - 0.203i)T + (2.25 + 12.8i)T^{2}
17 1+(0.714−1.96i)T+(−13.0−10.9i)T2 1 + (0.714 - 1.96i)T + (-13.0 - 10.9i)T^{2}
23 1+(2.05+1.72i)T+(3.99+22.6i)T2 1 + (2.05 + 1.72i)T + (3.99 + 22.6i)T^{2}
29 1+(−3.99−0.703i)T+(27.2+9.91i)T2 1 + (-3.99 - 0.703i)T + (27.2 + 9.91i)T^{2}
31 1+(−1.54−2.67i)T+(−15.5+26.8i)T2 1 + (-1.54 - 2.67i)T + (-15.5 + 26.8i)T^{2}
37 1+(−5.90+3.40i)T+(18.5−32.0i)T2 1 + (-5.90 + 3.40i)T + (18.5 - 32.0i)T^{2}
41 1+(0.706−0.592i)T+(7.11−40.3i)T2 1 + (0.706 - 0.592i)T + (7.11 - 40.3i)T^{2}
43 1+(−6.24−2.27i)T+(32.9+27.6i)T2 1 + (-6.24 - 2.27i)T + (32.9 + 27.6i)T^{2}
47 1+(1.14+3.13i)T+(−36.0+30.2i)T2 1 + (1.14 + 3.13i)T + (-36.0 + 30.2i)T^{2}
53 1+(−3.30−0.583i)T+(49.8+18.1i)T2 1 + (-3.30 - 0.583i)T + (49.8 + 18.1i)T^{2}
59 1+(−2.65−0.968i)T+(45.1+37.9i)T2 1 + (-2.65 - 0.968i)T + (45.1 + 37.9i)T^{2}
61 1+(4.20−5.01i)T+(−10.5−60.0i)T2 1 + (4.20 - 5.01i)T + (-10.5 - 60.0i)T^{2}
67 1+(2.11−2.51i)T+(−11.6−65.9i)T2 1 + (2.11 - 2.51i)T + (-11.6 - 65.9i)T^{2}
71 1+(4.60−12.6i)T+(−54.3−45.6i)T2 1 + (4.60 - 12.6i)T + (-54.3 - 45.6i)T^{2}
73 1+(4.27−0.754i)T+(68.5−24.9i)T2 1 + (4.27 - 0.754i)T + (68.5 - 24.9i)T^{2}
79 1+(−1.50+4.14i)T+(−60.5−50.7i)T2 1 + (-1.50 + 4.14i)T + (-60.5 - 50.7i)T^{2}
83 1+(4.75+2.74i)T+(41.5+71.8i)T2 1 + (4.75 + 2.74i)T + (41.5 + 71.8i)T^{2}
89 1+(1.83−10.3i)T+(−83.6−30.4i)T2 1 + (1.83 - 10.3i)T + (-83.6 - 30.4i)T^{2}
97 1+(−0.274−1.55i)T+(−91.1+33.1i)T2 1 + (-0.274 - 1.55i)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

−10.42175940706883792264450249064, −10.08289839993009933893530041402, −9.030567277553499824188771583319, −7.890421993999858765826335178171, −7.12271916953063460239419258758, −6.15105802450847708061221871018, −5.04943293040391798184501782820, −4.10117870801467737045561205187, −2.49252063317637271510752330857, −1.13412173912734363329494866366, 1.81543498355421478688908170145, 2.82349342585913581091197939517, 4.49918385040114631508057463357, 5.45158140162686789293876871673, 6.13852162723891706777829119555, 7.49423365837652732284535460832, 8.240407980696228809906253775706, 9.479818940332242559464980569337, 9.982683377798602126464191940760, 10.79574462249709777922731848443

Graph of the ZZ-function along the critical line