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Properties

Label 2-270-135.104-c2-0-12
Degree 22
Conductor 270270
Sign 0.832−0.554i0.832 - 0.554i
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  + (−1.32 − 0.483i)2-s + (−1.15 + 2.76i)3-s + (1.53 + 1.28i)4-s + (4.75 − 1.55i)5-s + (2.87 − 3.12i)6-s + (3.89 + 4.64i)7-s + (−1.41 − 2.44i)8-s + (−6.34 − 6.38i)9-s + (−7.06 − 0.234i)10-s + (19.9 − 3.51i)11-s + (−5.32 + 2.76i)12-s + (−4.33 − 11.8i)13-s + (−2.93 − 8.05i)14-s + (−1.17 + 14.9i)15-s + (0.694 + 3.93i)16-s + (−5.27 + 9.13i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.384 + 0.923i)3-s + (0.383 + 0.321i)4-s + (0.950 − 0.310i)5-s + (0.478 − 0.520i)6-s + (0.556 + 0.663i)7-s + (−0.176 − 0.306i)8-s + (−0.704 − 0.709i)9-s + (−0.706 − 0.0234i)10-s + (1.81 − 0.319i)11-s + (−0.443 + 0.230i)12-s + (−0.333 − 0.915i)13-s + (−0.209 − 0.575i)14-s + (−0.0782 + 0.996i)15-s + (0.0434 + 0.246i)16-s + (−0.310 + 0.537i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((0.832−0.554i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1) L(s)=((0.832−0.554i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: 0.832−0.554i0.832 - 0.554i
Analytic conductor: 7.356967.35696
Root analytic conductor: 2.712372.71237
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ270(239,⋅)\chi_{270} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1), 0.832−0.554i)(2,\ 270,\ (\ :1),\ 0.832 - 0.554i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.30209+0.393943i1.30209 + 0.393943i
L(12)L(\frac12) ≈\approx 1.30209+0.393943i1.30209 + 0.393943i
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+(1.32+0.483i)T 1 + (1.32 + 0.483i)T
3 1+(1.15−2.76i)T 1 + (1.15 - 2.76i)T
5 1+(−4.75+1.55i)T 1 + (-4.75 + 1.55i)T
good7 1+(−3.89−4.64i)T+(−8.50+48.2i)T2 1 + (-3.89 - 4.64i)T + (-8.50 + 48.2i)T^{2}
11 1+(−19.9+3.51i)T+(113.−41.3i)T2 1 + (-19.9 + 3.51i)T + (113. - 41.3i)T^{2}
13 1+(4.33+11.8i)T+(−129.+108.i)T2 1 + (4.33 + 11.8i)T + (-129. + 108. i)T^{2}
17 1+(5.27−9.13i)T+(−144.5−250.i)T2 1 + (5.27 - 9.13i)T + (-144.5 - 250. i)T^{2}
19 1+(5.65+9.80i)T+(−180.5+312.i)T2 1 + (5.65 + 9.80i)T + (-180.5 + 312. i)T^{2}
23 1+(−21.2−17.8i)T+(91.8+520.i)T2 1 + (-21.2 - 17.8i)T + (91.8 + 520. i)T^{2}
29 1+(7.25−19.9i)T+(−644.−540.i)T2 1 + (7.25 - 19.9i)T + (-644. - 540. i)T^{2}
31 1+(−14.4−12.1i)T+(166.+946.i)T2 1 + (-14.4 - 12.1i)T + (166. + 946. i)T^{2}
37 1+(−41.9−24.1i)T+(684.5+1.18e3i)T2 1 + (-41.9 - 24.1i)T + (684.5 + 1.18e3i)T^{2}
41 1+(−18.3−50.3i)T+(−1.28e3+1.08e3i)T2 1 + (-18.3 - 50.3i)T + (-1.28e3 + 1.08e3i)T^{2}
43 1+(60.0−10.5i)T+(1.73e3−632.i)T2 1 + (60.0 - 10.5i)T + (1.73e3 - 632. i)T^{2}
47 1+(−31.7+26.6i)T+(383.−2.17e3i)T2 1 + (-31.7 + 26.6i)T + (383. - 2.17e3i)T^{2}
53 1−46.1T+2.80e3T2 1 - 46.1T + 2.80e3T^{2}
59 1+(68.5+12.0i)T+(3.27e3+1.19e3i)T2 1 + (68.5 + 12.0i)T + (3.27e3 + 1.19e3i)T^{2}
61 1+(77.0−64.6i)T+(646.−3.66e3i)T2 1 + (77.0 - 64.6i)T + (646. - 3.66e3i)T^{2}
67 1+(30.8+84.7i)T+(−3.43e3+2.88e3i)T2 1 + (30.8 + 84.7i)T + (-3.43e3 + 2.88e3i)T^{2}
71 1+(−42.4−24.4i)T+(2.52e3+4.36e3i)T2 1 + (-42.4 - 24.4i)T + (2.52e3 + 4.36e3i)T^{2}
73 1+(−0.265+0.153i)T+(2.66e3−4.61e3i)T2 1 + (-0.265 + 0.153i)T + (2.66e3 - 4.61e3i)T^{2}
79 1+(82.8+30.1i)T+(4.78e3+4.01e3i)T2 1 + (82.8 + 30.1i)T + (4.78e3 + 4.01e3i)T^{2}
83 1+(−55.7−20.2i)T+(5.27e3+4.42e3i)T2 1 + (-55.7 - 20.2i)T + (5.27e3 + 4.42e3i)T^{2}
89 1+(34.8−20.1i)T+(3.96e3−6.85e3i)T2 1 + (34.8 - 20.1i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(78.3−13.8i)T+(8.84e3−3.21e3i)T2 1 + (78.3 - 13.8i)T + (8.84e3 - 3.21e3i)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

−11.54767665511213326631105193858, −10.78819798597915420217689367217, −9.749725026494129678308114400122, −9.089115130239797615239469420771, −8.445890976803012648003837445298, −6.63937480033453568111070538510, −5.73075143800885768191170118302, −4.61644273334610601265794432292, −3.05259876933408553002953569192, −1.31481530421987190591691176752, 1.15423993221391811784859033070, 2.20875045967289936975745636652, 4.49008940010538929011411191203, 6.02528781199774244031797844400, 6.76022189170106897236504412954, 7.42071161090931444074724934331, 8.808676474678541599679706569102, 9.559377771185604328765676205359, 10.75122669213724499580413880069, 11.48910357867911190912215653157

Graph of the ZZ-function along the critical line