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Properties

Label 2-405-45.43-c2-0-40
Degree 22
Conductor 405405
Sign −0.999−0.00266i-0.999 - 0.00266i
Analytic cond. 11.035411.0354
Root an. cond. 3.321963.32196
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  + (−0.578 − 2.15i)2-s + (−0.866 + 0.5i)4-s + (4.89 − 1.00i)5-s + (−1.83 − 6.83i)7-s + (−4.74 − 4.74i)8-s + (−5 − 10.0i)10-s + (−7.90 + 13.6i)11-s + (3.66 − 13.6i)13-s + (−13.6 + 7.90i)14-s + (−9.49 + 16.4i)16-s + (3.16 − 3.16i)17-s − 18i·19-s + (−3.74 + 3.31i)20-s + (34.1 + 9.15i)22-s + (1.15 − 4.31i)23-s + ⋯
L(s)  = 1  + (−0.289 − 1.07i)2-s + (−0.216 + 0.125i)4-s + (0.979 − 0.200i)5-s + (−0.261 − 0.975i)7-s + (−0.592 − 0.592i)8-s + (−0.5 − 1.00i)10-s + (−0.718 + 1.24i)11-s + (0.281 − 1.05i)13-s + (−0.978 + 0.564i)14-s + (−0.593 + 1.02i)16-s + (0.186 − 0.186i)17-s − 0.947i·19-s + (−0.187 + 0.165i)20-s + (1.55 + 0.415i)22-s + (0.0503 − 0.187i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.999−0.00266i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00266i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1) L(s)=((−0.999−0.00266i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.00266i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.999−0.00266i-0.999 - 0.00266i
Analytic conductor: 11.035411.0354
Root analytic conductor: 3.321963.32196
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ405(298,⋅)\chi_{405} (298, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1), −0.999−0.00266i)(2,\ 405,\ (\ :1),\ -0.999 - 0.00266i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.00185079+1.39142i0.00185079 + 1.39142i
L(12)L(\frac12) ≈\approx 0.00185079+1.39142i0.00185079 + 1.39142i
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 1
5 1+(−4.89+1.00i)T 1 + (-4.89 + 1.00i)T
good2 1+(0.578+2.15i)T+(−3.46+2i)T2 1 + (0.578 + 2.15i)T + (-3.46 + 2i)T^{2}
7 1+(1.83+6.83i)T+(−42.4+24.5i)T2 1 + (1.83 + 6.83i)T + (-42.4 + 24.5i)T^{2}
11 1+(7.90−13.6i)T+(−60.5−104.i)T2 1 + (7.90 - 13.6i)T + (-60.5 - 104. i)T^{2}
13 1+(−3.66+13.6i)T+(−146.−84.5i)T2 1 + (-3.66 + 13.6i)T + (-146. - 84.5i)T^{2}
17 1+(−3.16+3.16i)T−289iT2 1 + (-3.16 + 3.16i)T - 289iT^{2}
19 1+18iT−361T2 1 + 18iT - 361T^{2}
23 1+(−1.15+4.31i)T+(−458.−264.5i)T2 1 + (-1.15 + 4.31i)T + (-458. - 264.5i)T^{2}
29 1+(41.0+23.7i)T+(420.5+728.i)T2 1 + (41.0 + 23.7i)T + (420.5 + 728. i)T^{2}
31 1+(4+6.92i)T+(−480.5+832.i)T2 1 + (4 + 6.92i)T + (-480.5 + 832. i)T^{2}
37 1+(−10+10i)T−1.36e3iT2 1 + (-10 + 10i)T - 1.36e3iT^{2}
41 1+(−15.8−27.3i)T+(−840.5+1.45e3i)T2 1 + (-15.8 - 27.3i)T + (-840.5 + 1.45e3i)T^{2}
43 1+(13.6−3.66i)T+(1.60e3−924.5i)T2 1 + (13.6 - 3.66i)T + (1.60e3 - 924.5i)T^{2}
47 1+(−15.0−56.1i)T+(−1.91e3+1.10e3i)T2 1 + (-15.0 - 56.1i)T + (-1.91e3 + 1.10e3i)T^{2}
53 1+(25.2+25.2i)T+2.80e3iT2 1 + (25.2 + 25.2i)T + 2.80e3iT^{2}
59 1+(41.0−23.7i)T+(1.74e3−3.01e3i)T2 1 + (41.0 - 23.7i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−29+50.2i)T+(−1.86e3−3.22e3i)T2 1 + (-29 + 50.2i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(95.6+25.6i)T+(3.88e3+2.24e3i)T2 1 + (95.6 + 25.6i)T + (3.88e3 + 2.24e3i)T^{2}
71 1−63.2T+5.04e3T2 1 - 63.2T + 5.04e3T^{2}
73 1+(−55−55i)T+5.32e3iT2 1 + (-55 - 55i)T + 5.32e3iT^{2}
79 1+(−10.3−6i)T+(3.12e3+5.40e3i)T2 1 + (-10.3 - 6i)T + (3.12e3 + 5.40e3i)T^{2}
83 1+(−73.4+19.6i)T+(5.96e3−3.44e3i)T2 1 + (-73.4 + 19.6i)T + (5.96e3 - 3.44e3i)T^{2}
89 1−7.92e3T2 1 - 7.92e3T^{2}
97 1+(1.83+6.83i)T+(−8.14e3+4.70e3i)T2 1 + (1.83 + 6.83i)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

−10.56825604492436360903817127845, −9.800113668349454794141553387238, −9.349638864845290767118004740555, −7.83140248468933311229278226918, −6.84147411457202544339240173380, −5.71487241486456239995530710704, −4.46968934274527551023347130824, −3.04842711188112782948636152664, −2.00053678896078430869242214178, −0.62725646656248607109211676289, 2.05536650468224523762537964178, 3.28799782025716302389956245667, 5.39830571103905346020879529734, 5.84463633299566245071485931223, 6.63993391073045841483380632013, 7.77287222607639691664610541179, 8.822223606069609359750475273618, 9.214963629762623952012429592275, 10.50146890116556328365663550789, 11.43037919306720024692523330034

Graph of the ZZ-function along the critical line