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Properties

Label 2-270-45.43-c2-0-2
Degree 22
Conductor 270270
Sign −0.747−0.664i-0.747 - 0.664i
Analytic cond. 7.356967.35696
Root an. cond. 2.712372.71237
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  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (−1.44 − 4.78i)5-s + (1.61 + 6.01i)7-s + (−2 − 1.99i)8-s + (6.00 − 3.73i)10-s + (−5.39 + 9.34i)11-s + (−6.33 + 23.6i)13-s + (−7.63 + 4.40i)14-s + (1.99 − 3.46i)16-s + (11.1 − 11.1i)17-s + 20.2i·19-s + (7.29 + 6.83i)20-s + (−14.7 − 3.94i)22-s + (−7.01 + 26.1i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.250i)4-s + (−0.289 − 0.957i)5-s + (0.230 + 0.859i)7-s + (−0.250 − 0.249i)8-s + (0.600 − 0.373i)10-s + (−0.490 + 0.849i)11-s + (−0.487 + 1.81i)13-s + (−0.545 + 0.314i)14-s + (0.124 − 0.216i)16-s + (0.656 − 0.656i)17-s + 1.06i·19-s + (0.364 + 0.341i)20-s + (−0.669 − 0.179i)22-s + (−0.305 + 1.13i)23-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.747−0.664i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1) L(s)=((−0.747−0.664i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.747−0.664i-0.747 - 0.664i
Analytic conductor: 7.356967.35696
Root analytic conductor: 2.712372.71237
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ270(73,⋅)\chi_{270} (73, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1), −0.747−0.664i)(2,\ 270,\ (\ :1),\ -0.747 - 0.664i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.413343+1.08683i0.413343 + 1.08683i
L(12)L(\frac12) ≈\approx 0.413343+1.08683i0.413343 + 1.08683i
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)
bad2 1+(−0.366−1.36i)T 1 + (-0.366 - 1.36i)T
3 1 1
5 1+(1.44+4.78i)T 1 + (1.44 + 4.78i)T
good7 1+(−1.61−6.01i)T+(−42.4+24.5i)T2 1 + (-1.61 - 6.01i)T + (-42.4 + 24.5i)T^{2}
11 1+(5.39−9.34i)T+(−60.5−104.i)T2 1 + (5.39 - 9.34i)T + (-60.5 - 104. i)T^{2}
13 1+(6.33−23.6i)T+(−146.−84.5i)T2 1 + (6.33 - 23.6i)T + (-146. - 84.5i)T^{2}
17 1+(−11.1+11.1i)T−289iT2 1 + (-11.1 + 11.1i)T - 289iT^{2}
19 1−20.2iT−361T2 1 - 20.2iT - 361T^{2}
23 1+(7.01−26.1i)T+(−458.−264.5i)T2 1 + (7.01 - 26.1i)T + (-458. - 264.5i)T^{2}
29 1+(17.5+10.1i)T+(420.5+728.i)T2 1 + (17.5 + 10.1i)T + (420.5 + 728. i)T^{2}
31 1+(5.98+10.3i)T+(−480.5+832.i)T2 1 + (5.98 + 10.3i)T + (-480.5 + 832. i)T^{2}
37 1+(−40.1+40.1i)T−1.36e3iT2 1 + (-40.1 + 40.1i)T - 1.36e3iT^{2}
41 1+(−5.77−10.0i)T+(−840.5+1.45e3i)T2 1 + (-5.77 - 10.0i)T + (-840.5 + 1.45e3i)T^{2}
43 1+(18.9−5.07i)T+(1.60e3−924.5i)T2 1 + (18.9 - 5.07i)T + (1.60e3 - 924.5i)T^{2}
47 1+(−4.89−18.2i)T+(−1.91e3+1.10e3i)T2 1 + (-4.89 - 18.2i)T + (-1.91e3 + 1.10e3i)T^{2}
53 1+(31.5+31.5i)T+2.80e3iT2 1 + (31.5 + 31.5i)T + 2.80e3iT^{2}
59 1+(49.2−28.4i)T+(1.74e3−3.01e3i)T2 1 + (49.2 - 28.4i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(15.9−27.5i)T+(−1.86e3−3.22e3i)T2 1 + (15.9 - 27.5i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−79.8−21.3i)T+(3.88e3+2.24e3i)T2 1 + (-79.8 - 21.3i)T + (3.88e3 + 2.24e3i)T^{2}
71 1−22.4T+5.04e3T2 1 - 22.4T + 5.04e3T^{2}
73 1+(−11.7−11.7i)T+5.32e3iT2 1 + (-11.7 - 11.7i)T + 5.32e3iT^{2}
79 1+(−31.0−17.9i)T+(3.12e3+5.40e3i)T2 1 + (-31.0 - 17.9i)T + (3.12e3 + 5.40e3i)T^{2}
83 1+(−115.+30.9i)T+(5.96e3−3.44e3i)T2 1 + (-115. + 30.9i)T + (5.96e3 - 3.44e3i)T^{2}
89 1−12.2iT−7.92e3T2 1 - 12.2iT - 7.92e3T^{2}
97 1+(21.4+80.1i)T+(−8.14e3+4.70e3i)T2 1 + (21.4 + 80.1i)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

−12.10194663290119826769793118504, −11.55071238317207125851411281232, −9.620961689204282345416313918025, −9.280523934718833491406007439873, −8.025737549422648949701405484786, −7.33006333750385457696433177436, −5.87189455446492316598227274974, −4.99846624902091624430971444740, −4.02212183553168022052034100610, −1.94902672512416685216184452865, 0.55368832144223899003086666249, 2.72954624807512661199148186853, 3.58256881649687878036469093883, 5.00523345703243837724520209654, 6.23926068056757842523793235302, 7.58058769544714976393541803739, 8.269590628655325899073361737243, 9.867278829720148609462592136878, 10.74665700473149163727435393732, 10.91478541607051162053823335177

Graph of the ZZ-function along the critical line