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Properties

Label 2-150-5.4-c3-0-8
Degree 22
Conductor 150150
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 8.850288.85028
Root an. cond. 2.974942.97494
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  + 2i·2-s − 3i·3-s − 4·4-s + 6·6-s − 4i·7-s − 8i·8-s − 9·9-s − 48·11-s + 12i·12-s − 2i·13-s + 8·14-s + 16·16-s − 114i·17-s − 18i·18-s − 140·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.215i·7-s − 0.353i·8-s − 0.333·9-s − 1.31·11-s + 0.288i·12-s − 0.0426i·13-s + 0.152·14-s + 0.250·16-s − 1.62i·17-s − 0.235i·18-s − 1.69·19-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+3/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 8.850288.85028
Root analytic conductor: 2.974942.97494
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ150(49,⋅)\chi_{150} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :3/2), −0.447+0.894i)(2,\ 150,\ (\ :3/2),\ -0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 0.311319−0.503725i0.311319 - 0.503725i
L(12)L(\frac12) ≈\approx 0.311319−0.503725i0.311319 - 0.503725i
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 1−2iT 1 - 2iT
3 1+3iT 1 + 3iT
5 1 1
good7 1+4iT−343T2 1 + 4iT - 343T^{2}
11 1+48T+1.33e3T2 1 + 48T + 1.33e3T^{2}
13 1+2iT−2.19e3T2 1 + 2iT - 2.19e3T^{2}
17 1+114iT−4.91e3T2 1 + 114iT - 4.91e3T^{2}
19 1+140T+6.85e3T2 1 + 140T + 6.85e3T^{2}
23 1+72iT−1.21e4T2 1 + 72iT - 1.21e4T^{2}
29 1+210T+2.43e4T2 1 + 210T + 2.43e4T^{2}
31 1−272T+2.97e4T2 1 - 272T + 2.97e4T^{2}
37 1+334iT−5.06e4T2 1 + 334iT - 5.06e4T^{2}
41 1+198T+6.89e4T2 1 + 198T + 6.89e4T^{2}
43 1−268iT−7.95e4T2 1 - 268iT - 7.95e4T^{2}
47 1−216iT−1.03e5T2 1 - 216iT - 1.03e5T^{2}
53 1−78iT−1.48e5T2 1 - 78iT - 1.48e5T^{2}
59 1+240T+2.05e5T2 1 + 240T + 2.05e5T^{2}
61 1−302T+2.26e5T2 1 - 302T + 2.26e5T^{2}
67 1−596iT−3.00e5T2 1 - 596iT - 3.00e5T^{2}
71 1+768T+3.57e5T2 1 + 768T + 3.57e5T^{2}
73 1−478iT−3.89e5T2 1 - 478iT - 3.89e5T^{2}
79 1−640T+4.93e5T2 1 - 640T + 4.93e5T^{2}
83 1−348iT−5.71e5T2 1 - 348iT - 5.71e5T^{2}
89 1+210T+7.04e5T2 1 + 210T + 7.04e5T^{2}
97 1+1.53e3iT−9.12e5T2 1 + 1.53e3iT - 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

−12.56129776480021685096843606329, −11.21083133443429788769480921893, −10.12795762288647170570320878774, −8.823280012398518663406337505394, −7.82624810855910269559865874379, −6.96407135356038098304008064783, −5.76280917360728651188354811322, −4.54470384811990768391320096075, −2.58079993469889873831825000244, −0.26970457103126915749408202189, 2.15063714118031055240266891461, 3.64809476994345000178108283069, 4.89803569275988450126733203883, 6.13617407268324372371099619580, 8.008723332242600693263282145555, 8.815830045508566178127693547766, 10.22576905109508936003417735427, 10.59147494896833370313373735651, 11.78821021563569664780997133751, 12.86979543112994399977079057624

Graph of the ZZ-function along the critical line