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Properties

Label 2-270-135.103-c2-0-30
Degree 22
Conductor 270270
Sign −0.972+0.232i-0.972 + 0.232i
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 + (1.03 − 2.81i)3-s + (−1.96 − 0.347i)4-s + (3.02 + 3.97i)5-s + (−3.84 − 1.80i)6-s + (−1.93 − 1.35i)7-s + (−0.732 + 2.73i)8-s + (−6.86 − 5.82i)9-s + (5.97 − 3.77i)10-s + (−15.0 − 5.47i)11-s + (−3.01 + 5.18i)12-s + (−2.05 − 23.5i)13-s + (−2.14 + 2.56i)14-s + (14.3 − 4.41i)15-s + (3.75 + 1.36i)16-s + (−4.64 − 17.3i)17-s + ⋯
L(s)  = 1  + (0.0616 − 0.704i)2-s + (0.344 − 0.938i)3-s + (−0.492 − 0.0868i)4-s + (0.605 + 0.795i)5-s + (−0.640 − 0.300i)6-s + (−0.276 − 0.193i)7-s + (−0.0915 + 0.341i)8-s + (−0.762 − 0.646i)9-s + (0.597 − 0.377i)10-s + (−1.36 − 0.497i)11-s + (−0.251 + 0.432i)12-s + (−0.158 − 1.81i)13-s + (−0.153 + 0.182i)14-s + (0.955 − 0.294i)15-s + (0.234 + 0.0855i)16-s + (−0.273 − 1.02i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.972+0.232i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1) L(s)=((−0.972+0.232i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.972+0.232i-0.972 + 0.232i
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), −0.972+0.232i)(2,\ 270,\ (\ :1),\ -0.972 + 0.232i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.160641−1.36584i0.160641 - 1.36584i
L(12)L(\frac12) ≈\approx 0.160641−1.36584i0.160641 - 1.36584i
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+(−1.03+2.81i)T 1 + (-1.03 + 2.81i)T
5 1+(−3.02−3.97i)T 1 + (-3.02 - 3.97i)T
good7 1+(1.93+1.35i)T+(16.7+46.0i)T2 1 + (1.93 + 1.35i)T + (16.7 + 46.0i)T^{2}
11 1+(15.0+5.47i)T+(92.6+77.7i)T2 1 + (15.0 + 5.47i)T + (92.6 + 77.7i)T^{2}
13 1+(2.05+23.5i)T+(−166.+29.3i)T2 1 + (2.05 + 23.5i)T + (-166. + 29.3i)T^{2}
17 1+(4.64+17.3i)T+(−250.+144.5i)T2 1 + (4.64 + 17.3i)T + (-250. + 144.5i)T^{2}
19 1+(7.52−4.34i)T+(180.5−312.i)T2 1 + (7.52 - 4.34i)T + (180.5 - 312. i)T^{2}
23 1+(−30.4+21.3i)T+(180.−497.i)T2 1 + (-30.4 + 21.3i)T + (180. - 497. i)T^{2}
29 1+(−13.3−15.8i)T+(−146.+828.i)T2 1 + (-13.3 - 15.8i)T + (-146. + 828. i)T^{2}
31 1+(−3.91+22.1i)T+(−903.−328.i)T2 1 + (-3.91 + 22.1i)T + (-903. - 328. i)T^{2}
37 1+(−6.00−22.4i)T+(−1.18e3+684.5i)T2 1 + (-6.00 - 22.4i)T + (-1.18e3 + 684.5i)T^{2}
41 1+(−54.5−45.8i)T+(291.+1.65e3i)T2 1 + (-54.5 - 45.8i)T + (291. + 1.65e3i)T^{2}
43 1+(−6.76−14.5i)T+(−1.18e3+1.41e3i)T2 1 + (-6.76 - 14.5i)T + (-1.18e3 + 1.41e3i)T^{2}
47 1+(−16.8−11.8i)T+(755.+2.07e3i)T2 1 + (-16.8 - 11.8i)T + (755. + 2.07e3i)T^{2}
53 1+(−61.1−61.1i)T+2.80e3iT2 1 + (-61.1 - 61.1i)T + 2.80e3iT^{2}
59 1+(28.7+79.0i)T+(−2.66e3+2.23e3i)T2 1 + (28.7 + 79.0i)T + (-2.66e3 + 2.23e3i)T^{2}
61 1+(−3.63−20.6i)T+(−3.49e3+1.27e3i)T2 1 + (-3.63 - 20.6i)T + (-3.49e3 + 1.27e3i)T^{2}
67 1+(63.4−5.55i)T+(4.42e3−779.i)T2 1 + (63.4 - 5.55i)T + (4.42e3 - 779. i)T^{2}
71 1+(−57.9+100.i)T+(−2.52e3−4.36e3i)T2 1 + (-57.9 + 100. i)T + (-2.52e3 - 4.36e3i)T^{2}
73 1+(1.68−6.29i)T+(−4.61e3−2.66e3i)T2 1 + (1.68 - 6.29i)T + (-4.61e3 - 2.66e3i)T^{2}
79 1+(−14.2−16.9i)T+(−1.08e3+6.14e3i)T2 1 + (-14.2 - 16.9i)T + (-1.08e3 + 6.14e3i)T^{2}
83 1+(−74.9−6.56i)T+(6.78e3+1.19e3i)T2 1 + (-74.9 - 6.56i)T + (6.78e3 + 1.19e3i)T^{2}
89 1+(71.6−41.3i)T+(3.96e3−6.85e3i)T2 1 + (71.6 - 41.3i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(37.6−17.5i)T+(6.04e3−7.20e3i)T2 1 + (37.6 - 17.5i)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

−11.05606997422114418366178832419, −10.55237656156467445198880422316, −9.509795683131118300250699399506, −8.271381915619625797867139973442, −7.43438089299715238669682356888, −6.22673076555077289003565271205, −5.19717374648605339016053113119, −2.95446759842965716625680087850, −2.70240039571981072725535893730, −0.63519943790036431641456457416, 2.30345714369969988285798367040, 4.14033607127463246703566757501, 4.97048544504647038141942518244, 5.91143378846175976358995261047, 7.26999879085834554772086302685, 8.578725459580306296363922783094, 9.125010328174957886443411146310, 9.955707039959643264667313514690, 10.95202366659797408655883604761, 12.36764158764278479144167876592

Graph of the ZZ-function along the critical line