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Properties

Label 2-1620-1620.1519-c0-0-0
Degree 22
Conductor 16201620
Sign 0.925−0.378i0.925 - 0.378i
Analytic cond. 0.8084850.808485
Root an. cond. 0.8991580.899158
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.993 + 0.116i)2-s + (−0.0581 − 0.998i)3-s + (0.973 − 0.230i)4-s + (0.893 + 0.448i)5-s + (0.173 + 0.984i)6-s + (0.479 + 1.60i)7-s + (−0.939 + 0.342i)8-s + (−0.993 + 0.116i)9-s + (−0.939 − 0.342i)10-s + (−0.286 − 0.957i)12-s + (−0.661 − 1.53i)14-s + (0.396 − 0.918i)15-s + (0.893 − 0.448i)16-s + (0.973 − 0.230i)18-s + (0.973 + 0.230i)20-s + (1.57 − 0.571i)21-s + ⋯
L(s)  = 1  + (−0.993 + 0.116i)2-s + (−0.0581 − 0.998i)3-s + (0.973 − 0.230i)4-s + (0.893 + 0.448i)5-s + (0.173 + 0.984i)6-s + (0.479 + 1.60i)7-s + (−0.939 + 0.342i)8-s + (−0.993 + 0.116i)9-s + (−0.939 − 0.342i)10-s + (−0.286 − 0.957i)12-s + (−0.661 − 1.53i)14-s + (0.396 − 0.918i)15-s + (0.893 − 0.448i)16-s + (0.973 − 0.230i)18-s + (0.973 + 0.230i)20-s + (1.57 − 0.571i)21-s + ⋯

Functional equation

Λ(s)=(1620s/2 ΓC(s) L(s)=((0.925−0.378i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1620s/2 ΓC(s) L(s)=((0.925−0.378i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: 0.925−0.378i0.925 - 0.378i
Analytic conductor: 0.8084850.808485
Root analytic conductor: 0.8991580.899158
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1620(1519,⋅)\chi_{1620} (1519, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1620, ( :0), 0.925−0.378i)(2,\ 1620,\ (\ :0),\ 0.925 - 0.378i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.83877191670.8387719167
L(12)L(\frac12) ≈\approx 0.83877191670.8387719167
L(1)L(1) 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+(0.993−0.116i)T 1 + (0.993 - 0.116i)T
3 1+(0.0581+0.998i)T 1 + (0.0581 + 0.998i)T
5 1+(−0.893−0.448i)T 1 + (-0.893 - 0.448i)T
good7 1+(−0.479−1.60i)T+(−0.835+0.549i)T2 1 + (-0.479 - 1.60i)T + (-0.835 + 0.549i)T^{2}
11 1+(0.993+0.116i)T2 1 + (0.993 + 0.116i)T^{2}
13 1+(0.286+0.957i)T2 1 + (0.286 + 0.957i)T^{2}
17 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
19 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
23 1+(−0.164+0.549i)T+(−0.835−0.549i)T2 1 + (-0.164 + 0.549i)T + (-0.835 - 0.549i)T^{2}
29 1+(0.543−1.26i)T+(−0.686−0.727i)T2 1 + (0.543 - 1.26i)T + (-0.686 - 0.727i)T^{2}
31 1+(0.0581−0.998i)T2 1 + (0.0581 - 0.998i)T^{2}
37 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
41 1+(1.18+0.138i)T+(0.973+0.230i)T2 1 + (1.18 + 0.138i)T + (0.973 + 0.230i)T^{2}
43 1+(−1.57−1.03i)T+(0.396+0.918i)T2 1 + (-1.57 - 1.03i)T + (0.396 + 0.918i)T^{2}
47 1+(−1.36+1.44i)T+(−0.0581−0.998i)T2 1 + (-1.36 + 1.44i)T + (-0.0581 - 0.998i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.993−0.116i)T2 1 + (0.993 - 0.116i)T^{2}
61 1+(0.113+0.0268i)T+(0.893+0.448i)T2 1 + (0.113 + 0.0268i)T + (0.893 + 0.448i)T^{2}
67 1+(−0.313−0.727i)T+(−0.686+0.727i)T2 1 + (-0.313 - 0.727i)T + (-0.686 + 0.727i)T^{2}
71 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
73 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
79 1+(−0.973+0.230i)T2 1 + (-0.973 + 0.230i)T^{2}
83 1+(1.18−0.138i)T+(0.973−0.230i)T2 1 + (1.18 - 0.138i)T + (0.973 - 0.230i)T^{2}
89 1+(−1.86+0.679i)T+(0.766−0.642i)T2 1 + (-1.86 + 0.679i)T + (0.766 - 0.642i)T^{2}
97 1+(−0.597+0.802i)T2 1 + (-0.597 + 0.802i)T^{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

−9.288515830420581007252810186196, −8.866897027996998248081553287069, −8.190657214782065081332517978849, −7.22322072109484084811417718402, −6.56323847494876619928041096037, −5.73087478101002449731547211656, −5.32226192193410047024196593637, −2.97750638966527413938790887945, −2.30447945362343786712977226103, −1.52888332527984293629154593507, 0.976965450279633691849217498570, 2.30017651302046543405861657908, 3.61680962675085140229396312470, 4.44526357030582563300758680270, 5.49857881866288711541163620112, 6.34054731639104959735051972558, 7.38770941590066498154272597219, 8.066084132928010816223298114536, 9.034490373631036596082568588706, 9.549663450636930732467577529740

Graph of the ZZ-function along the critical line