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Properties

Label 2-2160-9.4-c1-0-7
Degree 22
Conductor 21602160
Sign 0.894−0.446i0.894 - 0.446i
Analytic cond. 17.247617.2476
Root an. cond. 4.153034.15303
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  + (0.5 − 0.866i)5-s + (−0.596 − 1.03i)7-s + (1.66 + 2.87i)11-s + (−0.853 + 1.47i)13-s + 6.34·17-s − 1.32·19-s + (−3.43 + 5.94i)23-s + (−0.499 − 0.866i)25-s + (1.01 + 1.75i)29-s + (−1.33 + 2.32i)31-s − 1.19·35-s + 3.32·37-s + (1.16 − 2.00i)41-s + (−3.17 − 5.49i)43-s + (6.38 + 11.0i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.225 − 0.390i)7-s + (0.500 + 0.867i)11-s + (−0.236 + 0.410i)13-s + 1.53·17-s − 0.303·19-s + (−0.715 + 1.23i)23-s + (−0.0999 − 0.173i)25-s + (0.188 + 0.326i)29-s + (−0.240 + 0.416i)31-s − 0.201·35-s + 0.545·37-s + (0.181 − 0.313i)41-s + (−0.484 − 0.838i)43-s + (0.931 + 1.61i)47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.894−0.446i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1/2) L(s)=((0.894−0.446i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.894−0.446i0.894 - 0.446i
Analytic conductor: 17.247617.2476
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2160(1441,⋅)\chi_{2160} (1441, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1/2), 0.894−0.446i)(2,\ 2160,\ (\ :1/2),\ 0.894 - 0.446i)

Particular Values

L(1)L(1) ≈\approx 1.8041061981.804106198
L(12)L(\frac12) ≈\approx 1.8041061981.804106198
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 1
3 1 1
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good7 1+(0.596+1.03i)T+(−3.5+6.06i)T2 1 + (0.596 + 1.03i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1.66−2.87i)T+(−5.5+9.52i)T2 1 + (-1.66 - 2.87i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.853−1.47i)T+(−6.5−11.2i)T2 1 + (0.853 - 1.47i)T + (-6.5 - 11.2i)T^{2}
17 1−6.34T+17T2 1 - 6.34T + 17T^{2}
19 1+1.32T+19T2 1 + 1.32T + 19T^{2}
23 1+(3.43−5.94i)T+(−11.5−19.9i)T2 1 + (3.43 - 5.94i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.01−1.75i)T+(−14.5+25.1i)T2 1 + (-1.01 - 1.75i)T + (-14.5 + 25.1i)T^{2}
31 1+(1.33−2.32i)T+(−15.5−26.8i)T2 1 + (1.33 - 2.32i)T + (-15.5 - 26.8i)T^{2}
37 1−3.32T+37T2 1 - 3.32T + 37T^{2}
41 1+(−1.16+2.00i)T+(−20.5−35.5i)T2 1 + (-1.16 + 2.00i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.17+5.49i)T+(−21.5+37.2i)T2 1 + (3.17 + 5.49i)T + (-21.5 + 37.2i)T^{2}
47 1+(−6.38−11.0i)T+(−23.5+40.7i)T2 1 + (-6.38 - 11.0i)T + (-23.5 + 40.7i)T^{2}
53 1+1.02T+53T2 1 + 1.02T + 53T^{2}
59 1+(−5.83+10.1i)T+(−29.5−51.0i)T2 1 + (-5.83 + 10.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(−4.86−8.43i)T+(−30.5+52.8i)T2 1 + (-4.86 - 8.43i)T + (-30.5 + 52.8i)T^{2}
67 1+(5.28−9.15i)T+(−33.5−58.0i)T2 1 + (5.28 - 9.15i)T + (-33.5 - 58.0i)T^{2}
71 1+1.06T+71T2 1 + 1.06T + 71T^{2}
73 1−14.0T+73T2 1 - 14.0T + 73T^{2}
79 1+(−0.707−1.22i)T+(−39.5+68.4i)T2 1 + (-0.707 - 1.22i)T + (-39.5 + 68.4i)T^{2}
83 1+(−5.91−10.2i)T+(−41.5+71.8i)T2 1 + (-5.91 - 10.2i)T + (-41.5 + 71.8i)T^{2}
89 1−11T+89T2 1 - 11T + 89T^{2}
97 1+(8.12+14.0i)T+(−48.5+84.0i)T2 1 + (8.12 + 14.0i)T + (-48.5 + 84.0i)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

−9.320843709612336085956147481328, −8.343570058849893374614580990139, −7.49583828929467784791213665325, −6.93205835666550962214286167965, −5.90826058924332888787569673674, −5.18861596850756816903715249126, −4.20019809706648510434635423272, −3.47955835569987155601008952572, −2.12873781012565151515449908239, −1.11068714305320948140990563365, 0.75158739770917954857393052295, 2.24470797477688030304503988336, 3.14731909944910594752448432653, 3.97919805202957668542783210547, 5.17034359175742150023689524102, 6.00242661752902940476086637384, 6.44833052970008064469814507689, 7.58735438516967775393006688362, 8.211074092540366874411247155727, 9.021448909766739809031774972743

Graph of the ZZ-function along the critical line