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Properties

Label 2-270-135.124-c1-0-11
Degree 22
Conductor 270270
Sign −0.181+0.983i-0.181 + 0.983i
Analytic cond. 2.155962.15596
Root an. cond. 1.468311.46831
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.726 − 1.57i)3-s + (−0.173 − 0.984i)4-s + (1.96 − 1.06i)5-s + (−1.67 − 0.454i)6-s + (4.83 + 0.853i)7-s + (−0.866 − 0.500i)8-s + (−1.94 + 2.28i)9-s + (0.445 − 2.19i)10-s + (−1.60 + 0.584i)11-s + (−1.42 + 0.988i)12-s + (−1.95 − 2.32i)13-s + (3.76 − 3.15i)14-s + (−3.10 − 2.31i)15-s + (−0.939 + 0.342i)16-s + (−3.04 + 1.75i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.419 − 0.907i)3-s + (−0.0868 − 0.492i)4-s + (0.878 − 0.477i)5-s + (−0.682 − 0.185i)6-s + (1.82 + 0.322i)7-s + (−0.306 − 0.176i)8-s + (−0.648 + 0.761i)9-s + (0.140 − 0.692i)10-s + (−0.484 + 0.176i)11-s + (−0.410 + 0.285i)12-s + (−0.541 − 0.645i)13-s + (1.00 − 0.843i)14-s + (−0.801 − 0.597i)15-s + (−0.234 + 0.0855i)16-s + (−0.737 + 0.425i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.181+0.983i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((−0.181+0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.181+0.983i-0.181 + 0.983i
Analytic conductor: 2.155962.15596
Root analytic conductor: 1.468311.46831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ270(259,⋅)\chi_{270} (259, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1/2), −0.181+0.983i)(2,\ 270,\ (\ :1/2),\ -0.181 + 0.983i)

Particular Values

L(1)L(1) ≈\approx 1.08351−1.30154i1.08351 - 1.30154i
L(12)L(\frac12) ≈\approx 1.08351−1.30154i1.08351 - 1.30154i
L(32)L(\frac{3}{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+(−0.642+0.766i)T 1 + (-0.642 + 0.766i)T
3 1+(0.726+1.57i)T 1 + (0.726 + 1.57i)T
5 1+(−1.96+1.06i)T 1 + (-1.96 + 1.06i)T
good7 1+(−4.83−0.853i)T+(6.57+2.39i)T2 1 + (-4.83 - 0.853i)T + (6.57 + 2.39i)T^{2}
11 1+(1.60−0.584i)T+(8.42−7.07i)T2 1 + (1.60 - 0.584i)T + (8.42 - 7.07i)T^{2}
13 1+(1.95+2.32i)T+(−2.25+12.8i)T2 1 + (1.95 + 2.32i)T + (-2.25 + 12.8i)T^{2}
17 1+(3.04−1.75i)T+(8.5−14.7i)T2 1 + (3.04 - 1.75i)T + (8.5 - 14.7i)T^{2}
19 1+(0.936−1.62i)T+(−9.5−16.4i)T2 1 + (0.936 - 1.62i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.79−0.492i)T+(21.6−7.86i)T2 1 + (2.79 - 0.492i)T + (21.6 - 7.86i)T^{2}
29 1+(−2.46−2.06i)T+(5.03+28.5i)T2 1 + (-2.46 - 2.06i)T + (5.03 + 28.5i)T^{2}
31 1+(−0.994−5.64i)T+(−29.1+10.6i)T2 1 + (-0.994 - 5.64i)T + (-29.1 + 10.6i)T^{2}
37 1+(−1.92+1.11i)T+(18.5−32.0i)T2 1 + (-1.92 + 1.11i)T + (18.5 - 32.0i)T^{2}
41 1+(0.451−0.379i)T+(7.11−40.3i)T2 1 + (0.451 - 0.379i)T + (7.11 - 40.3i)T^{2}
43 1+(−3.99−10.9i)T+(−32.9+27.6i)T2 1 + (-3.99 - 10.9i)T + (-32.9 + 27.6i)T^{2}
47 1+(13.0+2.30i)T+(44.1+16.0i)T2 1 + (13.0 + 2.30i)T + (44.1 + 16.0i)T^{2}
53 1+13.5iT−53T2 1 + 13.5iT - 53T^{2}
59 1+(−4.14−1.50i)T+(45.1+37.9i)T2 1 + (-4.14 - 1.50i)T + (45.1 + 37.9i)T^{2}
61 1+(−1.58+8.98i)T+(−57.3−20.8i)T2 1 + (-1.58 + 8.98i)T + (-57.3 - 20.8i)T^{2}
67 1+(−4.47−5.32i)T+(−11.6+65.9i)T2 1 + (-4.47 - 5.32i)T + (-11.6 + 65.9i)T^{2}
71 1+(−2.54−4.40i)T+(−35.5+61.4i)T2 1 + (-2.54 - 4.40i)T + (-35.5 + 61.4i)T^{2}
73 1+(11.3+6.52i)T+(36.5+63.2i)T2 1 + (11.3 + 6.52i)T + (36.5 + 63.2i)T^{2}
79 1+(−4.76−3.99i)T+(13.7+77.7i)T2 1 + (-4.76 - 3.99i)T + (13.7 + 77.7i)T^{2}
83 1+(−2.47+2.94i)T+(−14.4−81.7i)T2 1 + (-2.47 + 2.94i)T + (-14.4 - 81.7i)T^{2}
89 1+(0.280−0.485i)T+(−44.5−77.0i)T2 1 + (0.280 - 0.485i)T + (-44.5 - 77.0i)T^{2}
97 1+(−2.45−6.75i)T+(−74.3+62.3i)T2 1 + (-2.45 - 6.75i)T + (-74.3 + 62.3i)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

−11.74681969949951846884947748109, −10.96079420562879778980714507673, −10.07401859483918537573211906946, −8.557447589210433716741693146032, −7.918275396450432170962768751214, −6.41838858677418896838976120219, −5.28506840135424408175060368550, −4.81217061360378970995451661530, −2.37876783880269770556079623381, −1.50013571180351682427847651803, 2.38750218519510380805570780994, 4.28892028757515149407215648500, 4.97568058997454717248750180844, 5.94517965415082109422250890286, 7.12684662808173551610852020582, 8.304412464106792180210164281979, 9.350465171528011574602884771641, 10.45795191394607768829103496869, 11.20901384389406478858217828923, 11.92678138025859801739322230710

Graph of the ZZ-function along the critical line