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Properties

Label 2-252-4.3-c2-0-27
Degree 22
Conductor 252252
Sign 0.722+0.691i0.722 + 0.691i
Analytic cond. 6.866506.86650
Root an. cond. 2.620402.62040
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  + (1.83 − 0.785i)2-s + (2.76 − 2.88i)4-s + 8.17·5-s − 2.64i·7-s + (2.81 − 7.48i)8-s + (15.0 − 6.42i)10-s + 15.5i·11-s − 20.4·13-s + (−2.07 − 4.86i)14-s + (−0.701 − 15.9i)16-s + 5.97·17-s + 4.19i·19-s + (22.6 − 23.6i)20-s + (12.2 + 28.6i)22-s − 29.3i·23-s + ⋯
L(s)  = 1  + (0.919 − 0.392i)2-s + (0.691 − 0.722i)4-s + 1.63·5-s − 0.377i·7-s + (0.352 − 0.935i)8-s + (1.50 − 0.642i)10-s + 1.41i·11-s − 1.57·13-s + (−0.148 − 0.347i)14-s + (−0.0438 − 0.999i)16-s + 0.351·17-s + 0.220i·19-s + (1.13 − 1.18i)20-s + (0.555 + 1.30i)22-s − 1.27i·23-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.722+0.691i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+1) L(s)=((0.722+0.691i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.722+0.691i0.722 + 0.691i
Analytic conductor: 6.866506.86650
Root analytic conductor: 2.620402.62040
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ252(127,⋅)\chi_{252} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :1), 0.722+0.691i)(2,\ 252,\ (\ :1),\ 0.722 + 0.691i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 3.06393−1.22995i3.06393 - 1.22995i
L(12)L(\frac12) ≈\approx 3.06393−1.22995i3.06393 - 1.22995i
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+(−1.83+0.785i)T 1 + (-1.83 + 0.785i)T
3 1 1
7 1+2.64iT 1 + 2.64iT
good5 1−8.17T+25T2 1 - 8.17T + 25T^{2}
11 1−15.5iT−121T2 1 - 15.5iT - 121T^{2}
13 1+20.4T+169T2 1 + 20.4T + 169T^{2}
17 1−5.97T+289T2 1 - 5.97T + 289T^{2}
19 1−4.19iT−361T2 1 - 4.19iT - 361T^{2}
23 1+29.3iT−529T2 1 + 29.3iT - 529T^{2}
29 1−7.89T+841T2 1 - 7.89T + 841T^{2}
31 1−35.3iT−961T2 1 - 35.3iT - 961T^{2}
37 1+49.2T+1.36e3T2 1 + 49.2T + 1.36e3T^{2}
41 1−4.11T+1.68e3T2 1 - 4.11T + 1.68e3T^{2}
43 1+19.6iT−1.84e3T2 1 + 19.6iT - 1.84e3T^{2}
47 1−30.5iT−2.20e3T2 1 - 30.5iT - 2.20e3T^{2}
53 1+40.4T+2.80e3T2 1 + 40.4T + 2.80e3T^{2}
59 1−83.1iT−3.48e3T2 1 - 83.1iT - 3.48e3T^{2}
61 1+3.43T+3.72e3T2 1 + 3.43T + 3.72e3T^{2}
67 1−5.17iT−4.48e3T2 1 - 5.17iT - 4.48e3T^{2}
71 1−75.1iT−5.04e3T2 1 - 75.1iT - 5.04e3T^{2}
73 1+21.5T+5.32e3T2 1 + 21.5T + 5.32e3T^{2}
79 1+62.5iT−6.24e3T2 1 + 62.5iT - 6.24e3T^{2}
83 1+58.0iT−6.88e3T2 1 + 58.0iT - 6.88e3T^{2}
89 1−135.T+7.92e3T2 1 - 135.T + 7.92e3T^{2}
97 1−61.8T+9.40e3T2 1 - 61.8T + 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

−12.15759297109257409928593022812, −10.36307462835332048903475528870, −10.21754760240950786992181232411, −9.250034145365656606602484700655, −7.29725769954609331194488266267, −6.53153707235630932694330255300, −5.29207035504288866703062729583, −4.57703950084335610532838597755, −2.69715197689494351950952293759, −1.72766111432829072856611742054, 2.08014835690906896639222213825, 3.20733304763936678510245947722, 5.07252529131239786191039343046, 5.68236344952853509400372102353, 6.57441874043086571169584177347, 7.82152967330396910321356740947, 9.088225108272317651196500603991, 9.995250905036845663186283358901, 11.16221582404342950263131417787, 12.14347625567671311092363223065

Graph of the ZZ-function along the critical line