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Properties

Label 2-48-3.2-c2-0-0
Degree 22
Conductor 4848
Sign 0.333−0.942i0.333 - 0.942i
Analytic cond. 1.307901.30790
Root an. cond. 1.143631.14363
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 2.82i)3-s + 5.65i·5-s + 6·7-s + (−7.00 − 5.65i)9-s − 5.65i·11-s + 10·13-s + (−16.0 − 5.65i)15-s − 22.6i·17-s − 2·19-s + (−6 + 16.9i)21-s + 11.3i·23-s − 7.00·25-s + (23.0 − 14.1i)27-s + 16.9i·29-s + 22·31-s + ⋯
L(s)  = 1  + (−0.333 + 0.942i)3-s + 1.13i·5-s + 0.857·7-s + (−0.777 − 0.628i)9-s − 0.514i·11-s + 0.769·13-s + (−1.06 − 0.377i)15-s − 1.33i·17-s − 0.105·19-s + (−0.285 + 0.808i)21-s + 0.491i·23-s − 0.280·25-s + (0.851 − 0.523i)27-s + 0.585i·29-s + 0.709·31-s + ⋯

Functional equation

Λ(s)=(48s/2 ΓC(s) L(s)=((0.333−0.942i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(48s/2 ΓC(s+1) L(s)=((0.333−0.942i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4848    =    24⋅32^{4} \cdot 3
Sign: 0.333−0.942i0.333 - 0.942i
Analytic conductor: 1.307901.30790
Root analytic conductor: 1.143631.14363
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ48(17,⋅)\chi_{48} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 48, ( :1), 0.333−0.942i)(2,\ 48,\ (\ :1),\ 0.333 - 0.942i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.867701+0.613557i0.867701 + 0.613557i
L(12)L(\frac12) ≈\approx 0.867701+0.613557i0.867701 + 0.613557i
L(2)L(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
3 1+(1−2.82i)T 1 + (1 - 2.82i)T
good5 1−5.65iT−25T2 1 - 5.65iT - 25T^{2}
7 1−6T+49T2 1 - 6T + 49T^{2}
11 1+5.65iT−121T2 1 + 5.65iT - 121T^{2}
13 1−10T+169T2 1 - 10T + 169T^{2}
17 1+22.6iT−289T2 1 + 22.6iT - 289T^{2}
19 1+2T+361T2 1 + 2T + 361T^{2}
23 1−11.3iT−529T2 1 - 11.3iT - 529T^{2}
29 1−16.9iT−841T2 1 - 16.9iT - 841T^{2}
31 1−22T+961T2 1 - 22T + 961T^{2}
37 1+6T+1.36e3T2 1 + 6T + 1.36e3T^{2}
41 1−33.9iT−1.68e3T2 1 - 33.9iT - 1.68e3T^{2}
43 1+82T+1.84e3T2 1 + 82T + 1.84e3T^{2}
47 1+67.8iT−2.20e3T2 1 + 67.8iT - 2.20e3T^{2}
53 1+62.2iT−2.80e3T2 1 + 62.2iT - 2.80e3T^{2}
59 1+73.5iT−3.48e3T2 1 + 73.5iT - 3.48e3T^{2}
61 1+86T+3.72e3T2 1 + 86T + 3.72e3T^{2}
67 1+2T+4.48e3T2 1 + 2T + 4.48e3T^{2}
71 1−124.iT−5.04e3T2 1 - 124. iT - 5.04e3T^{2}
73 1−82T+5.32e3T2 1 - 82T + 5.32e3T^{2}
79 1+10T+6.24e3T2 1 + 10T + 6.24e3T^{2}
83 1−73.5iT−6.88e3T2 1 - 73.5iT - 6.88e3T^{2}
89 1+33.9iT−7.92e3T2 1 + 33.9iT - 7.92e3T^{2}
97 1+94T+9.40e3T2 1 + 94T + 9.40e3T^{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

−15.57043223853227462553880449452, −14.63570113284640684849674349504, −13.74779223154144646161891469990, −11.59147277940091212263005169755, −11.06628828728236644478383214993, −9.903901932416096870875403707062, −8.409602321260687544304099349977, −6.66243153714466825535656970955, −5.10001899190853530745711742341, −3.29890576054193911342403457106, 1.51516477393485154652426285439, 4.69649591590057934138421818576, 6.15665602481661924731916389292, 7.898390187659691057364673114554, 8.733397303057901341913019031213, 10.69952286346587240119431300380, 11.97297937000266713434486219270, 12.79875184868878539502629179602, 13.80983076217042636039033038115, 15.18205616010965058745776826752

Graph of the ZZ-function along the critical line