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Properties

Label 2-360-120.53-c1-0-9
Degree 22
Conductor 360360
Sign 0.251−0.967i0.251 - 0.967i
Analytic cond. 2.874612.87461
Root an. cond. 1.695461.69546
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  + (−0.473 + 1.33i)2-s + (−1.55 − 1.26i)4-s + (1.45 + 1.69i)5-s + (1.53 − 1.53i)7-s + (2.41 − 1.47i)8-s + (−2.95 + 1.13i)10-s + 2.72·11-s + (0.857 − 0.857i)13-s + (1.31 + 2.76i)14-s + (0.818 + 3.91i)16-s + (2.55 + 2.55i)17-s − 3.54·19-s + (−0.113 − 4.47i)20-s + (−1.28 + 3.63i)22-s + (−0.626 + 0.626i)23-s + ⋯
L(s)  = 1  + (−0.334 + 0.942i)2-s + (−0.776 − 0.630i)4-s + (0.650 + 0.759i)5-s + (0.578 − 0.578i)7-s + (0.853 − 0.520i)8-s + (−0.933 + 0.358i)10-s + 0.821·11-s + (0.237 − 0.237i)13-s + (0.351 + 0.738i)14-s + (0.204 + 0.978i)16-s + (0.619 + 0.619i)17-s − 0.812·19-s + (−0.0254 − 0.999i)20-s + (−0.274 + 0.774i)22-s + (−0.130 + 0.130i)23-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((0.251−0.967i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+1/2) L(s)=((0.251−0.967i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: 0.251−0.967i0.251 - 0.967i
Analytic conductor: 2.874612.87461
Root analytic conductor: 1.695461.69546
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ360(53,⋅)\chi_{360} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :1/2), 0.251−0.967i)(2,\ 360,\ (\ :1/2),\ 0.251 - 0.967i)

Particular Values

L(1)L(1) ≈\approx 1.02457+0.792700i1.02457 + 0.792700i
L(12)L(\frac12) ≈\approx 1.02457+0.792700i1.02457 + 0.792700i
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+(0.473−1.33i)T 1 + (0.473 - 1.33i)T
3 1 1
5 1+(−1.45−1.69i)T 1 + (-1.45 - 1.69i)T
good7 1+(−1.53+1.53i)T−7iT2 1 + (-1.53 + 1.53i)T - 7iT^{2}
11 1−2.72T+11T2 1 - 2.72T + 11T^{2}
13 1+(−0.857+0.857i)T−13iT2 1 + (-0.857 + 0.857i)T - 13iT^{2}
17 1+(−2.55−2.55i)T+17iT2 1 + (-2.55 - 2.55i)T + 17iT^{2}
19 1+3.54T+19T2 1 + 3.54T + 19T^{2}
23 1+(0.626−0.626i)T−23iT2 1 + (0.626 - 0.626i)T - 23iT^{2}
29 1−5.12iT−29T2 1 - 5.12iT - 29T^{2}
31 1−7.89T+31T2 1 - 7.89T + 31T^{2}
37 1+(4.21+4.21i)T+37iT2 1 + (4.21 + 4.21i)T + 37iT^{2}
41 1−12.4iT−41T2 1 - 12.4iT - 41T^{2}
43 1+(−5.67+5.67i)T−43iT2 1 + (-5.67 + 5.67i)T - 43iT^{2}
47 1+(9.45+9.45i)T+47iT2 1 + (9.45 + 9.45i)T + 47iT^{2}
53 1+(6.46+6.46i)T+53iT2 1 + (6.46 + 6.46i)T + 53iT^{2}
59 1+2.51iT−59T2 1 + 2.51iT - 59T^{2}
61 1+9.49iT−61T2 1 + 9.49iT - 61T^{2}
67 1+(−9.91−9.91i)T+67iT2 1 + (-9.91 - 9.91i)T + 67iT^{2}
71 1−2.19iT−71T2 1 - 2.19iT - 71T^{2}
73 1+(5.71+5.71i)T+73iT2 1 + (5.71 + 5.71i)T + 73iT^{2}
79 1+12.7iT−79T2 1 + 12.7iT - 79T^{2}
83 1+(3.58+3.58i)T+83iT2 1 + (3.58 + 3.58i)T + 83iT^{2}
89 1−10.2T+89T2 1 - 10.2T + 89T^{2}
97 1+(1.29−1.29i)T−97iT2 1 + (1.29 - 1.29i)T - 97iT^{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

−11.34007864886512833715379418446, −10.46118404073064157889278062136, −9.830492173143899820861584712913, −8.699681524118870143119804935410, −7.82985019775255639138346080098, −6.76969275564290286518626552317, −6.14150629474063518942628290237, −4.92999810818858339485702278479, −3.65340844576073601755125590738, −1.54072316143711982322900013436, 1.28270719605535487965185863854, 2.49959852062694329510674496872, 4.15148735461421931864076358009, 5.07809591274727682486585633844, 6.29976758612026043986589884068, 7.933277793186658596398595302111, 8.722423709933537238997330812200, 9.415395914773666729719334960886, 10.24560549379053070702314506218, 11.39607754721079929269210229242

Graph of the ZZ-function along the critical line