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Properties

Label 2-60e2-20.3-c1-0-11
Degree 22
Conductor 36003600
Sign −0.525−0.850i-0.525 - 0.850i
Analytic cond. 28.746128.7461
Root an. cond. 5.361545.36154
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  + (2.82 + 2.82i)7-s + 5.65i·11-s + (−3 − 3i)13-s + (−1 + i)17-s + 5.65·19-s + (−2.82 + 2.82i)23-s + 4i·29-s + (−5 + 5i)37-s + (2.82 − 2.82i)43-s + (−2.82 − 2.82i)47-s + 9.00i·49-s + (1 + i)53-s − 11.3·59-s + 4·61-s + (−2.82 − 2.82i)67-s + ⋯
L(s)  = 1  + (1.06 + 1.06i)7-s + 1.70i·11-s + (−0.832 − 0.832i)13-s + (−0.242 + 0.242i)17-s + 1.29·19-s + (−0.589 + 0.589i)23-s + 0.742i·29-s + (−0.821 + 0.821i)37-s + (0.431 − 0.431i)43-s + (−0.412 − 0.412i)47-s + 1.28i·49-s + (0.137 + 0.137i)53-s − 1.47·59-s + 0.512·61-s + (−0.345 − 0.345i)67-s + ⋯

Functional equation

Λ(s)=(3600s/2 ΓC(s) L(s)=((−0.525−0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3600s/2 ΓC(s+1/2) L(s)=((−0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36003600    =    24⋅32⋅522^{4} \cdot 3^{2} \cdot 5^{2}
Sign: −0.525−0.850i-0.525 - 0.850i
Analytic conductor: 28.746128.7461
Root analytic conductor: 5.361545.36154
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3600(2143,⋅)\chi_{3600} (2143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3600, ( :1/2), −0.525−0.850i)(2,\ 3600,\ (\ :1/2),\ -0.525 - 0.850i)

Particular Values

L(1)L(1) ≈\approx 1.5978278681.597827868
L(12)L(\frac12) ≈\approx 1.5978278681.597827868
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
3 1 1
5 1 1
good7 1+(−2.82−2.82i)T+7iT2 1 + (-2.82 - 2.82i)T + 7iT^{2}
11 1−5.65iT−11T2 1 - 5.65iT - 11T^{2}
13 1+(3+3i)T+13iT2 1 + (3 + 3i)T + 13iT^{2}
17 1+(1−i)T−17iT2 1 + (1 - i)T - 17iT^{2}
19 1−5.65T+19T2 1 - 5.65T + 19T^{2}
23 1+(2.82−2.82i)T−23iT2 1 + (2.82 - 2.82i)T - 23iT^{2}
29 1−4iT−29T2 1 - 4iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+(5−5i)T−37iT2 1 + (5 - 5i)T - 37iT^{2}
41 1+41T2 1 + 41T^{2}
43 1+(−2.82+2.82i)T−43iT2 1 + (-2.82 + 2.82i)T - 43iT^{2}
47 1+(2.82+2.82i)T+47iT2 1 + (2.82 + 2.82i)T + 47iT^{2}
53 1+(−1−i)T+53iT2 1 + (-1 - i)T + 53iT^{2}
59 1+11.3T+59T2 1 + 11.3T + 59T^{2}
61 1−4T+61T2 1 - 4T + 61T^{2}
67 1+(2.82+2.82i)T+67iT2 1 + (2.82 + 2.82i)T + 67iT^{2}
71 1+5.65iT−71T2 1 + 5.65iT - 71T^{2}
73 1+(−3−3i)T+73iT2 1 + (-3 - 3i)T + 73iT^{2}
79 1−5.65T+79T2 1 - 5.65T + 79T^{2}
83 1+(2.82−2.82i)T−83iT2 1 + (2.82 - 2.82i)T - 83iT^{2}
89 1−8iT−89T2 1 - 8iT - 89T^{2}
97 1+(−3+3i)T−97iT2 1 + (-3 + 3i)T - 97iT^{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

−8.794796025728683514956336449650, −7.913047109868366266505926922030, −7.51390359981155042970980433307, −6.68655004300170128442774677577, −5.45738358289832092741758954516, −5.17197427931218091411101764151, −4.41517846647434849297529089030, −3.18315848910153165069586832434, −2.22214724967517208015401005097, −1.52282754867929191619145003511, 0.46744966646273437289178753980, 1.51035261332147738061972477188, 2.69443330031777719240047533335, 3.71648414277240351807893849025, 4.44931284907403082260903561480, 5.20762519778478018989846180889, 6.04255652033969541337234493063, 6.94392528987117401848515753958, 7.64257309698909325788279014951, 8.167728715279014914564746515498

Graph of the ZZ-function along the critical line