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Properties

Label 2-252-12.11-c3-0-3
Degree 22
Conductor 252252
Sign −0.666−0.745i-0.666 - 0.745i
Analytic cond. 14.868414.8684
Root an. cond. 3.855963.85596
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.88 − 2.11i)2-s + (−0.909 − 7.94i)4-s + 21.8i·5-s − 7i·7-s + (−18.4 − 13.0i)8-s + (46.1 + 41.1i)10-s − 66.1·11-s − 61.5·13-s + (−14.7 − 13.1i)14-s + (−62.3 + 14.4i)16-s + 11.7i·17-s − 87.9i·19-s + (173. − 19.8i)20-s + (−124. + 139. i)22-s + 13.2·23-s + ⋯
L(s)  = 1  + (0.665 − 0.746i)2-s + (−0.113 − 0.993i)4-s + 1.95i·5-s − 0.377i·7-s + (−0.817 − 0.576i)8-s + (1.45 + 1.30i)10-s − 1.81·11-s − 1.31·13-s + (−0.282 − 0.251i)14-s + (−0.974 + 0.225i)16-s + 0.167i·17-s − 1.06i·19-s + (1.94 − 0.222i)20-s + (−1.20 + 1.35i)22-s + 0.119·23-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((−0.666−0.745i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+3/2) L(s)=((−0.666−0.745i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: −0.666−0.745i-0.666 - 0.745i
Analytic conductor: 14.868414.8684
Root analytic conductor: 3.855963.85596
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ252(71,⋅)\chi_{252} (71, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :3/2), −0.666−0.745i)(2,\ 252,\ (\ :3/2),\ -0.666 - 0.745i)

Particular Values

L(2)L(2) ≈\approx 0.34046084600.3404608460
L(12)L(\frac12) ≈\approx 0.34046084600.3404608460
L(52)L(\frac{5}{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+(−1.88+2.11i)T 1 + (-1.88 + 2.11i)T
3 1 1
7 1+7iT 1 + 7iT
good5 1−21.8iT−125T2 1 - 21.8iT - 125T^{2}
11 1+66.1T+1.33e3T2 1 + 66.1T + 1.33e3T^{2}
13 1+61.5T+2.19e3T2 1 + 61.5T + 2.19e3T^{2}
17 1−11.7iT−4.91e3T2 1 - 11.7iT - 4.91e3T^{2}
19 1+87.9iT−6.85e3T2 1 + 87.9iT - 6.85e3T^{2}
23 1−13.2T+1.21e4T2 1 - 13.2T + 1.21e4T^{2}
29 1+7.53iT−2.43e4T2 1 + 7.53iT - 2.43e4T^{2}
31 1−45.1iT−2.97e4T2 1 - 45.1iT - 2.97e4T^{2}
37 1−207.T+5.06e4T2 1 - 207.T + 5.06e4T^{2}
41 1−172.iT−6.89e4T2 1 - 172. iT - 6.89e4T^{2}
43 1−331.iT−7.95e4T2 1 - 331. iT - 7.95e4T^{2}
47 1−436.T+1.03e5T2 1 - 436.T + 1.03e5T^{2}
53 1−282.iT−1.48e5T2 1 - 282. iT - 1.48e5T^{2}
59 1−553.T+2.05e5T2 1 - 553.T + 2.05e5T^{2}
61 1+262.T+2.26e5T2 1 + 262.T + 2.26e5T^{2}
67 1−89.8iT−3.00e5T2 1 - 89.8iT - 3.00e5T^{2}
71 1+891.T+3.57e5T2 1 + 891.T + 3.57e5T^{2}
73 1+506.T+3.89e5T2 1 + 506.T + 3.89e5T^{2}
79 1+836.iT−4.93e5T2 1 + 836. iT - 4.93e5T^{2}
83 1+114.T+5.71e5T2 1 + 114.T + 5.71e5T^{2}
89 1−162.iT−7.04e5T2 1 - 162. iT - 7.04e5T^{2}
97 1+1.05e3T+9.12e5T2 1 + 1.05e3T + 9.12e5T^{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

−11.76449443404940317943011568741, −10.86919758774321137399942166954, −10.41207204312902088108037994370, −9.640247144248555948136589237036, −7.67884293894270456633856474014, −6.93710035470479005338524637818, −5.75182809248377210820779995854, −4.51070108107426160528074439583, −2.92459240640868214941759809736, −2.52063851231095441804275411816, 0.097213382828791341623580477369, 2.39380043906675767039569567797, 4.23359262839044893939915801818, 5.24431923516112754360711505211, 5.59519442804877853441623718517, 7.49075484420588953575666994591, 8.139184812939871322125595882609, 9.015291516746877123049740864654, 10.07777230356818397390368593218, 11.81459841283539351300633986002

Graph of the ZZ-function along the critical line