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Properties

Label 2-45-45.7-c2-0-9
Degree 22
Conductor 4545
Sign −0.972+0.231i-0.972 + 0.231i
Analytic cond. 1.226161.22616
Root an. cond. 1.107321.10732
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  + (−0.891 + 0.238i)2-s + (−2.93 − 0.613i)3-s + (−2.72 + 1.57i)4-s + (−1.31 − 4.82i)5-s + (2.76 − 0.154i)6-s + (−11.0 + 2.96i)7-s + (4.66 − 4.66i)8-s + (8.24 + 3.60i)9-s + (2.32 + 3.98i)10-s + (−1.30 + 2.26i)11-s + (8.97 − 2.94i)12-s + (2.85 + 0.764i)13-s + (9.15 − 5.28i)14-s + (0.908 + 14.9i)15-s + (3.24 − 5.62i)16-s + (−13.6 − 13.6i)17-s + ⋯
L(s)  = 1  + (−0.445 + 0.119i)2-s + (−0.978 − 0.204i)3-s + (−0.681 + 0.393i)4-s + (−0.263 − 0.964i)5-s + (0.460 − 0.0257i)6-s + (−1.57 + 0.423i)7-s + (0.583 − 0.583i)8-s + (0.916 + 0.400i)9-s + (0.232 + 0.398i)10-s + (−0.118 + 0.205i)11-s + (0.747 − 0.245i)12-s + (0.219 + 0.0588i)13-s + (0.653 − 0.377i)14-s + (0.0605 + 0.998i)15-s + (0.203 − 0.351i)16-s + (−0.805 − 0.805i)17-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((−0.972+0.231i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+1) L(s)=((−0.972+0.231i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: −0.972+0.231i-0.972 + 0.231i
Analytic conductor: 1.226161.22616
Root analytic conductor: 1.107321.10732
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ45(7,⋅)\chi_{45} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :1), −0.972+0.231i)(2,\ 45,\ (\ :1),\ -0.972 + 0.231i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.00883755−0.0753363i0.00883755 - 0.0753363i
L(12)L(\frac12) ≈\approx 0.00883755−0.0753363i0.00883755 - 0.0753363i
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)
bad3 1+(2.93+0.613i)T 1 + (2.93 + 0.613i)T
5 1+(1.31+4.82i)T 1 + (1.31 + 4.82i)T
good2 1+(0.891−0.238i)T+(3.46−2i)T2 1 + (0.891 - 0.238i)T + (3.46 - 2i)T^{2}
7 1+(11.0−2.96i)T+(42.4−24.5i)T2 1 + (11.0 - 2.96i)T + (42.4 - 24.5i)T^{2}
11 1+(1.30−2.26i)T+(−60.5−104.i)T2 1 + (1.30 - 2.26i)T + (-60.5 - 104. i)T^{2}
13 1+(−2.85−0.764i)T+(146.+84.5i)T2 1 + (-2.85 - 0.764i)T + (146. + 84.5i)T^{2}
17 1+(13.6+13.6i)T+289iT2 1 + (13.6 + 13.6i)T + 289iT^{2}
19 1−11.4iT−361T2 1 - 11.4iT - 361T^{2}
23 1+(10.7+2.89i)T+(458.+264.5i)T2 1 + (10.7 + 2.89i)T + (458. + 264.5i)T^{2}
29 1+(23.0+13.2i)T+(420.5+728.i)T2 1 + (23.0 + 13.2i)T + (420.5 + 728. i)T^{2}
31 1+(21.8+37.8i)T+(−480.5+832.i)T2 1 + (21.8 + 37.8i)T + (-480.5 + 832. i)T^{2}
37 1+(−14.4−14.4i)T+1.36e3iT2 1 + (-14.4 - 14.4i)T + 1.36e3iT^{2}
41 1+(0.924+1.60i)T+(−840.5+1.45e3i)T2 1 + (0.924 + 1.60i)T + (-840.5 + 1.45e3i)T^{2}
43 1+(−13.1−49.0i)T+(−1.60e3+924.5i)T2 1 + (-13.1 - 49.0i)T + (-1.60e3 + 924.5i)T^{2}
47 1+(−61.3+16.4i)T+(1.91e3−1.10e3i)T2 1 + (-61.3 + 16.4i)T + (1.91e3 - 1.10e3i)T^{2}
53 1+(−6.43+6.43i)T−2.80e3iT2 1 + (-6.43 + 6.43i)T - 2.80e3iT^{2}
59 1+(49.5−28.5i)T+(1.74e3−3.01e3i)T2 1 + (49.5 - 28.5i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−16.8+29.1i)T+(−1.86e3−3.22e3i)T2 1 + (-16.8 + 29.1i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(8.41−31.4i)T+(−3.88e3−2.24e3i)T2 1 + (8.41 - 31.4i)T + (-3.88e3 - 2.24e3i)T^{2}
71 1+63.3T+5.04e3T2 1 + 63.3T + 5.04e3T^{2}
73 1+(50.9−50.9i)T−5.32e3iT2 1 + (50.9 - 50.9i)T - 5.32e3iT^{2}
79 1+(59.5+34.4i)T+(3.12e3+5.40e3i)T2 1 + (59.5 + 34.4i)T + (3.12e3 + 5.40e3i)T^{2}
83 1+(27.2+101.i)T+(−5.96e3+3.44e3i)T2 1 + (27.2 + 101. i)T + (-5.96e3 + 3.44e3i)T^{2}
89 1+136.iT−7.92e3T2 1 + 136. iT - 7.92e3T^{2}
97 1+(−35.3+9.45i)T+(8.14e3−4.70e3i)T2 1 + (-35.3 + 9.45i)T + (8.14e3 - 4.70e3i)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

−15.74506975849466771794885377177, −13.28964233527661404395910286129, −12.83989734527563251362243609728, −11.77228005436347436422326954664, −9.922131869829286878968921186388, −9.075346681453765638725233321055, −7.48261849523490948154929272899, −5.88671328211566458818444045397, −4.23608471815439681462766068793, −0.10344797455491693729038421039, 3.88970192032924855507853808204, 5.91356579067965384402021214356, 7.07967646450504475361302308778, 9.195417246657395744438245237521, 10.36004510580348200770618132911, 10.92580357236198921807755371520, 12.65778505027315745170098916918, 13.71458643830547044720790113801, 15.24174971228115613056315835615, 16.22764442891380521289493301971

Graph of the ZZ-function along the critical line