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Properties

Label 2-270-135.103-c2-0-5
Degree 22
Conductor 270270
Sign −0.911+0.410i-0.911 + 0.410i
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  + (−0.123 + 1.40i)2-s + (0.0747 + 2.99i)3-s + (−1.96 − 0.347i)4-s + (−1.43 + 4.78i)5-s + (−4.23 − 0.264i)6-s + (5.12 + 3.58i)7-s + (0.732 − 2.73i)8-s + (−8.98 + 0.448i)9-s + (−6.57 − 2.61i)10-s + (1.10 + 0.400i)11-s + (0.894 − 5.93i)12-s + (1.98 + 22.7i)13-s + (−5.68 + 6.77i)14-s + (−14.4 − 3.94i)15-s + (3.75 + 1.36i)16-s + (−7.82 − 29.2i)17-s + ⋯
L(s)  = 1  + (−0.0616 + 0.704i)2-s + (0.0249 + 0.999i)3-s + (−0.492 − 0.0868i)4-s + (−0.286 + 0.957i)5-s + (−0.705 − 0.0440i)6-s + (0.732 + 0.512i)7-s + (0.0915 − 0.341i)8-s + (−0.998 + 0.0498i)9-s + (−0.657 − 0.261i)10-s + (0.100 + 0.0364i)11-s + (0.0745 − 0.494i)12-s + (0.152 + 1.74i)13-s + (−0.406 + 0.484i)14-s + (−0.964 − 0.263i)15-s + (0.234 + 0.0855i)16-s + (−0.460 − 1.71i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.911+0.410i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1) L(s)=((−0.911+0.410i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.911+0.410i-0.911 + 0.410i
Analytic conductor: 7.356967.35696
Root analytic conductor: 2.712372.71237
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ270(103,⋅)\chi_{270} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1), −0.911+0.410i)(2,\ 270,\ (\ :1),\ -0.911 + 0.410i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.250352−1.16670i0.250352 - 1.16670i
L(12)L(\frac12) ≈\approx 0.250352−1.16670i0.250352 - 1.16670i
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+(0.123−1.40i)T 1 + (0.123 - 1.40i)T
3 1+(−0.0747−2.99i)T 1 + (-0.0747 - 2.99i)T
5 1+(1.43−4.78i)T 1 + (1.43 - 4.78i)T
good7 1+(−5.12−3.58i)T+(16.7+46.0i)T2 1 + (-5.12 - 3.58i)T + (16.7 + 46.0i)T^{2}
11 1+(−1.10−0.400i)T+(92.6+77.7i)T2 1 + (-1.10 - 0.400i)T + (92.6 + 77.7i)T^{2}
13 1+(−1.98−22.7i)T+(−166.+29.3i)T2 1 + (-1.98 - 22.7i)T + (-166. + 29.3i)T^{2}
17 1+(7.82+29.2i)T+(−250.+144.5i)T2 1 + (7.82 + 29.2i)T + (-250. + 144.5i)T^{2}
19 1+(10.4−6.04i)T+(180.5−312.i)T2 1 + (10.4 - 6.04i)T + (180.5 - 312. i)T^{2}
23 1+(−23.0+16.1i)T+(180.−497.i)T2 1 + (-23.0 + 16.1i)T + (180. - 497. i)T^{2}
29 1+(−2.86−3.41i)T+(−146.+828.i)T2 1 + (-2.86 - 3.41i)T + (-146. + 828. i)T^{2}
31 1+(0.0505−0.286i)T+(−903.−328.i)T2 1 + (0.0505 - 0.286i)T + (-903. - 328. i)T^{2}
37 1+(−7.76−28.9i)T+(−1.18e3+684.5i)T2 1 + (-7.76 - 28.9i)T + (-1.18e3 + 684.5i)T^{2}
41 1+(−1.89−1.59i)T+(291.+1.65e3i)T2 1 + (-1.89 - 1.59i)T + (291. + 1.65e3i)T^{2}
43 1+(−7.84−16.8i)T+(−1.18e3+1.41e3i)T2 1 + (-7.84 - 16.8i)T + (-1.18e3 + 1.41e3i)T^{2}
47 1+(18.2+12.8i)T+(755.+2.07e3i)T2 1 + (18.2 + 12.8i)T + (755. + 2.07e3i)T^{2}
53 1+(−23.0−23.0i)T+2.80e3iT2 1 + (-23.0 - 23.0i)T + 2.80e3iT^{2}
59 1+(−3.86−10.6i)T+(−2.66e3+2.23e3i)T2 1 + (-3.86 - 10.6i)T + (-2.66e3 + 2.23e3i)T^{2}
61 1+(−16.9−95.8i)T+(−3.49e3+1.27e3i)T2 1 + (-16.9 - 95.8i)T + (-3.49e3 + 1.27e3i)T^{2}
67 1+(34.3−3.00i)T+(4.42e3−779.i)T2 1 + (34.3 - 3.00i)T + (4.42e3 - 779. i)T^{2}
71 1+(61.7−106.i)T+(−2.52e3−4.36e3i)T2 1 + (61.7 - 106. i)T + (-2.52e3 - 4.36e3i)T^{2}
73 1+(5.98−22.3i)T+(−4.61e3−2.66e3i)T2 1 + (5.98 - 22.3i)T + (-4.61e3 - 2.66e3i)T^{2}
79 1+(−83.6−99.7i)T+(−1.08e3+6.14e3i)T2 1 + (-83.6 - 99.7i)T + (-1.08e3 + 6.14e3i)T^{2}
83 1+(82.2+7.19i)T+(6.78e3+1.19e3i)T2 1 + (82.2 + 7.19i)T + (6.78e3 + 1.19e3i)T^{2}
89 1+(−127.+73.4i)T+(3.96e3−6.85e3i)T2 1 + (-127. + 73.4i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(−129.+60.5i)T+(6.04e3−7.20e3i)T2 1 + (-129. + 60.5i)T + (6.04e3 - 7.20e3i)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

−11.72084312816644931812243204822, −11.39600302477591259781723264692, −10.26463098537988787329956642499, −9.198364647120445750243615480775, −8.568333836374881332343213344348, −7.22340837585173517346979376940, −6.35403893622229856923221246786, −4.98204580861885280308882538676, −4.17467262365150285532928609736, −2.61385307467046467842375584861, 0.63580576899667318868658721070, 1.77811708925330248449176507792, 3.50650325352715344277742997286, 4.87958794195230152958175679175, 5.98467216150745149617728577248, 7.62331250515850925594731336785, 8.218458385991456670579651196039, 9.014983330415650678851601392350, 10.57634480134736290629164902883, 11.16951940951390502900120512292

Graph of the ZZ-function along the critical line