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Properties

Label 2-33e2-99.47-c0-0-0
Degree 22
Conductor 10891089
Sign 0.913+0.406i0.913 + 0.406i
Analytic cond. 0.5434810.543481
Root an. cond. 0.7372120.737212
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  + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−1.72 − 0.181i)5-s + (−0.104 − 0.994i)9-s + (0.5 − 0.866i)12-s + (1.28 − 1.15i)15-s + (0.913 − 0.406i)16-s + (1.72 − 0.181i)20-s + (1.95 + 0.415i)25-s + (0.809 + 0.587i)27-s + (0.913 + 0.406i)31-s + (0.309 + 0.951i)36-s + (0.309 − 0.951i)37-s + 1.73i·45-s + (0.360 − 1.69i)47-s + (−0.309 + 0.951i)48-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−1.72 − 0.181i)5-s + (−0.104 − 0.994i)9-s + (0.5 − 0.866i)12-s + (1.28 − 1.15i)15-s + (0.913 − 0.406i)16-s + (1.72 − 0.181i)20-s + (1.95 + 0.415i)25-s + (0.809 + 0.587i)27-s + (0.913 + 0.406i)31-s + (0.309 + 0.951i)36-s + (0.309 − 0.951i)37-s + 1.73i·45-s + (0.360 − 1.69i)47-s + (−0.309 + 0.951i)48-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((0.913+0.406i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s) L(s)=((0.913+0.406i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: 0.913+0.406i0.913 + 0.406i
Analytic conductor: 0.5434810.543481
Root analytic conductor: 0.7372120.737212
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1089(245,⋅)\chi_{1089} (245, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1089, ( :0), 0.913+0.406i)(2,\ 1089,\ (\ :0),\ 0.913 + 0.406i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.36356562760.3635656276
L(12)L(\frac12) ≈\approx 0.36356562760.3635656276
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)
bad3 1+(0.669−0.743i)T 1 + (0.669 - 0.743i)T
11 1 1
good2 1+(0.978−0.207i)T2 1 + (0.978 - 0.207i)T^{2}
5 1+(1.72+0.181i)T+(0.978+0.207i)T2 1 + (1.72 + 0.181i)T + (0.978 + 0.207i)T^{2}
7 1+(−0.104+0.994i)T2 1 + (-0.104 + 0.994i)T^{2}
13 1+(0.669+0.743i)T2 1 + (0.669 + 0.743i)T^{2}
17 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
19 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(0.104−0.994i)T2 1 + (0.104 - 0.994i)T^{2}
31 1+(−0.913−0.406i)T+(0.669+0.743i)T2 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2}
37 1+(−0.309+0.951i)T+(−0.809−0.587i)T2 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}
41 1+(0.104+0.994i)T2 1 + (0.104 + 0.994i)T^{2}
43 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
47 1+(−0.360+1.69i)T+(−0.913−0.406i)T2 1 + (-0.360 + 1.69i)T + (-0.913 - 0.406i)T^{2}
53 1+(1.01+1.40i)T+(−0.309+0.951i)T2 1 + (1.01 + 1.40i)T + (-0.309 + 0.951i)T^{2}
59 1+(−0.360−1.69i)T+(−0.913+0.406i)T2 1 + (-0.360 - 1.69i)T + (-0.913 + 0.406i)T^{2}
61 1+(0.669−0.743i)T2 1 + (0.669 - 0.743i)T^{2}
67 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+(−1.01+1.40i)T+(−0.309−0.951i)T2 1 + (-1.01 + 1.40i)T + (-0.309 - 0.951i)T^{2}
73 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
79 1+(−0.978+0.207i)T2 1 + (-0.978 + 0.207i)T^{2}
83 1+(−0.669+0.743i)T2 1 + (-0.669 + 0.743i)T^{2}
89 1−T2 1 - T^{2}
97 1+(−0.104−0.994i)T+(−0.978+0.207i)T2 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)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

−10.06020877222711465274330322461, −9.094337691612032780875889808841, −8.465960088777360389020179576863, −7.69963777593424874400648462560, −6.70676376674748128068529941457, −5.39683063960763899127126602799, −4.66703828281066826279713335114, −3.95840125991140156828474424085, −3.31919641074968655133768504355, −0.51327809106746081110577102323, 0.990170039254317290133442846089, 2.99356484526563916301213786187, 4.22374297474166463179812140824, 4.76203928270803249437902214337, 5.91608957379477845132158476649, 6.84993963502090522039560678051, 7.900531962492548836387086529612, 8.077287584725451346216435465038, 9.186002337863880186572645181543, 10.28095163521730226145548995752

Graph of the ZZ-function along the critical line