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Properties

Label 2-2160-15.14-c2-0-23
Degree 22
Conductor 21602160
Sign −0.948−0.316i-0.948 - 0.316i
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  + (−1.58 + 4.74i)5-s + 3i·7-s + 9.48i·11-s + 21i·13-s − 12.6·17-s + 31·19-s + 22.1·23-s + (−20 − 15.0i)25-s + 47.4i·29-s + 16·31-s + (−14.2 − 4.74i)35-s + 27i·37-s − 47.4i·41-s + 48i·43-s + 12.6·47-s + ⋯
L(s)  = 1  + (−0.316 + 0.948i)5-s + 0.428i·7-s + 0.862i·11-s + 1.61i·13-s − 0.744·17-s + 1.63·19-s + 0.962·23-s + (−0.800 − 0.600i)25-s + 1.63i·29-s + 0.516·31-s + (−0.406 − 0.135i)35-s + 0.729i·37-s − 1.15i·41-s + 1.11i·43-s + 0.269·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.948−0.316i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((−0.948−0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.948−0.316i-0.948 - 0.316i
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), −0.948−0.316i)(2,\ 2160,\ (\ :1),\ -0.948 - 0.316i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.5758044091.575804409
L(12)L(\frac12) ≈\approx 1.5758044091.575804409
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+(1.58−4.74i)T 1 + (1.58 - 4.74i)T
good7 1−3iT−49T2 1 - 3iT - 49T^{2}
11 1−9.48iT−121T2 1 - 9.48iT - 121T^{2}
13 1−21iT−169T2 1 - 21iT - 169T^{2}
17 1+12.6T+289T2 1 + 12.6T + 289T^{2}
19 1−31T+361T2 1 - 31T + 361T^{2}
23 1−22.1T+529T2 1 - 22.1T + 529T^{2}
29 1−47.4iT−841T2 1 - 47.4iT - 841T^{2}
31 1−16T+961T2 1 - 16T + 961T^{2}
37 1−27iT−1.36e3T2 1 - 27iT - 1.36e3T^{2}
41 1+47.4iT−1.68e3T2 1 + 47.4iT - 1.68e3T^{2}
43 1−48iT−1.84e3T2 1 - 48iT - 1.84e3T^{2}
47 1−12.6T+2.20e3T2 1 - 12.6T + 2.20e3T^{2}
53 1+41.1T+2.80e3T2 1 + 41.1T + 2.80e3T^{2}
59 1+37.9iT−3.48e3T2 1 + 37.9iT - 3.48e3T^{2}
61 1+T+3.72e3T2 1 + T + 3.72e3T^{2}
67 1+21iT−4.48e3T2 1 + 21iT - 4.48e3T^{2}
71 1+28.4iT−5.04e3T2 1 + 28.4iT - 5.04e3T^{2}
73 1−27iT−5.32e3T2 1 - 27iT - 5.32e3T^{2}
79 1−T+6.24e3T2 1 - T + 6.24e3T^{2}
83 1+110.T+6.88e3T2 1 + 110.T + 6.88e3T^{2}
89 1−113.iT−7.92e3T2 1 - 113. iT - 7.92e3T^{2}
97 1+93iT−9.40e3T2 1 + 93iT - 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

−9.299517670251271803437503627346, −8.567824488023899396229593337206, −7.38554573052155404741237600961, −7.01900723288827215981748115446, −6.33786976491996886054093974411, −5.16241608849150930528975852097, −4.42180453147960649836156957675, −3.38837375895475496290333927180, −2.52404370638662269982798470188, −1.48630706301953191401225376768, 0.45428692620403345115813374308, 1.07858715357387718607577657506, 2.72853579326341594798068331471, 3.58020344766317732482899897263, 4.53155380859506497815264655145, 5.40048490831536428858916961246, 5.94708946174985058045013498214, 7.21061448928144323311936943483, 7.85385253735654769566826614630, 8.471604528785469544818426363408

Graph of the ZZ-function along the critical line