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Properties

Label 2-2160-15.14-c2-0-14
Degree 22
Conductor 21602160
Sign −0.938+0.344i-0.938 + 0.344i
Analytic cond. 58.855758.8557
Root an. cond. 7.671747.67174
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  + (1.72 + 4.69i)5-s + 13.6i·7-s − 1.59i·11-s − 17.2i·13-s − 21.0·17-s − 11.6·19-s + 24.4·23-s + (−19.0 + 16.1i)25-s + 40.5i·29-s + 38.6·31-s + (−63.9 + 23.4i)35-s + 32.5i·37-s − 5.41i·41-s + 15.7i·43-s − 32.8·47-s + ⋯
L(s)  = 1  + (0.344 + 0.938i)5-s + 1.94i·7-s − 0.145i·11-s − 1.32i·13-s − 1.24·17-s − 0.615·19-s + 1.06·23-s + (−0.763 + 0.646i)25-s + 1.39i·29-s + 1.24·31-s + (−1.82 + 0.669i)35-s + 0.878i·37-s − 0.132i·41-s + 0.366i·43-s − 0.699·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.938+0.344i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((−0.938+0.344i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.938+0.344i-0.938 + 0.344i
Analytic conductor: 58.855758.8557
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2160(1889,⋅)\chi_{2160} (1889, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1), −0.938+0.344i)(2,\ 2160,\ (\ :1),\ -0.938 + 0.344i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.89701274240.8970127424
L(12)L(\frac12) ≈\approx 0.89701274240.8970127424
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 1
3 1 1
5 1+(−1.72−4.69i)T 1 + (-1.72 - 4.69i)T
good7 1−13.6iT−49T2 1 - 13.6iT - 49T^{2}
11 1+1.59iT−121T2 1 + 1.59iT - 121T^{2}
13 1+17.2iT−169T2 1 + 17.2iT - 169T^{2}
17 1+21.0T+289T2 1 + 21.0T + 289T^{2}
19 1+11.6T+361T2 1 + 11.6T + 361T^{2}
23 1−24.4T+529T2 1 - 24.4T + 529T^{2}
29 1−40.5iT−841T2 1 - 40.5iT - 841T^{2}
31 1−38.6T+961T2 1 - 38.6T + 961T^{2}
37 1−32.5iT−1.36e3T2 1 - 32.5iT - 1.36e3T^{2}
41 1+5.41iT−1.68e3T2 1 + 5.41iT - 1.68e3T^{2}
43 1−15.7iT−1.84e3T2 1 - 15.7iT - 1.84e3T^{2}
47 1+32.8T+2.20e3T2 1 + 32.8T + 2.20e3T^{2}
53 1+97.5T+2.80e3T2 1 + 97.5T + 2.80e3T^{2}
59 1−88.1iT−3.48e3T2 1 - 88.1iT - 3.48e3T^{2}
61 1−9.07T+3.72e3T2 1 - 9.07T + 3.72e3T^{2}
67 1−26.8iT−4.48e3T2 1 - 26.8iT - 4.48e3T^{2}
71 1+109.iT−5.04e3T2 1 + 109. iT - 5.04e3T^{2}
73 1−29.0iT−5.32e3T2 1 - 29.0iT - 5.32e3T^{2}
79 1−18.3T+6.24e3T2 1 - 18.3T + 6.24e3T^{2}
83 1−23.3T+6.88e3T2 1 - 23.3T + 6.88e3T^{2}
89 1+147.iT−7.92e3T2 1 + 147. iT - 7.92e3T^{2}
97 1+100.iT−9.40e3T2 1 + 100. iT - 9.40e3T^{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

−9.188855053650386917443128429552, −8.637464845452145050076050810591, −7.920278646650604340860208182277, −6.73782930433166524889037215054, −6.23885404797119453082454120520, −5.45927499812976104113171371489, −4.71886769812606290526577530013, −2.99213926233855199102116318820, −2.88129624976147687425269647416, −1.72124426408465343531404646503, 0.21771617033843884307215372333, 1.23613496872309019393755888201, 2.23898273341210482037387059605, 3.83816392420179023034218063001, 4.41624612295896372386217563813, 4.91676880651828804844616861613, 6.46404905683351356913554260992, 6.69621638300269335875447824162, 7.74588954598707374575834267513, 8.430950185498737223992945286378

Graph of the ZZ-function along the critical line