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Properties

Label 2-2160-15.14-c2-0-70
Degree 22
Conductor 21602160
Sign 0.425+0.905i0.425 + 0.905i
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  + (4.52 − 2.12i)5-s − 0.343i·7-s + 3.68i·11-s − 5.50i·13-s + 6.36·17-s + 2.62·19-s − 12.2·23-s + (15.9 − 19.2i)25-s − 24.0i·29-s + 1.04·31-s + (−0.729 − 1.55i)35-s − 36.2i·37-s + 8.06i·41-s + 19.3i·43-s + 38.7·47-s + ⋯
L(s)  = 1  + (0.905 − 0.425i)5-s − 0.0490i·7-s + 0.334i·11-s − 0.423i·13-s + 0.374·17-s + 0.138·19-s − 0.533·23-s + (0.638 − 0.769i)25-s − 0.828i·29-s + 0.0337·31-s + (−0.0208 − 0.0443i)35-s − 0.979i·37-s + 0.196i·41-s + 0.449i·43-s + 0.824·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.425+0.905i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((0.425+0.905i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.425+0.905i0.425 + 0.905i
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), 0.425+0.905i)(2,\ 2160,\ (\ :1),\ 0.425 + 0.905i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.3396625922.339662592
L(12)L(\frac12) ≈\approx 2.3396625922.339662592
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+(−4.52+2.12i)T 1 + (-4.52 + 2.12i)T
good7 1+0.343iT−49T2 1 + 0.343iT - 49T^{2}
11 1−3.68iT−121T2 1 - 3.68iT - 121T^{2}
13 1+5.50iT−169T2 1 + 5.50iT - 169T^{2}
17 1−6.36T+289T2 1 - 6.36T + 289T^{2}
19 1−2.62T+361T2 1 - 2.62T + 361T^{2}
23 1+12.2T+529T2 1 + 12.2T + 529T^{2}
29 1+24.0iT−841T2 1 + 24.0iT - 841T^{2}
31 1−1.04T+961T2 1 - 1.04T + 961T^{2}
37 1+36.2iT−1.36e3T2 1 + 36.2iT - 1.36e3T^{2}
41 1−8.06iT−1.68e3T2 1 - 8.06iT - 1.68e3T^{2}
43 1−19.3iT−1.84e3T2 1 - 19.3iT - 1.84e3T^{2}
47 1−38.7T+2.20e3T2 1 - 38.7T + 2.20e3T^{2}
53 1−3.24T+2.80e3T2 1 - 3.24T + 2.80e3T^{2}
59 1+51.5iT−3.48e3T2 1 + 51.5iT - 3.48e3T^{2}
61 1−28.7T+3.72e3T2 1 - 28.7T + 3.72e3T^{2}
67 1−77.8iT−4.48e3T2 1 - 77.8iT - 4.48e3T^{2}
71 1+87.5iT−5.04e3T2 1 + 87.5iT - 5.04e3T^{2}
73 1+108.iT−5.32e3T2 1 + 108. iT - 5.32e3T^{2}
79 1+78.1T+6.24e3T2 1 + 78.1T + 6.24e3T^{2}
83 1−62.1T+6.88e3T2 1 - 62.1T + 6.88e3T^{2}
89 1+35.2iT−7.92e3T2 1 + 35.2iT - 7.92e3T^{2}
97 1+65.0iT−9.40e3T2 1 + 65.0iT - 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

−8.849519825282976006781134323337, −7.996194158688117391799364227753, −7.25514227567987444353365206764, −6.21792264960395147356071660227, −5.65691641120240140700293742054, −4.81939800031024998323921807132, −3.89665815126146577486180656671, −2.69657455309781269778047474724, −1.80147428834616529069564419399, −0.62059141102165334455613256438, 1.12940376088264433790456759508, 2.20327428346451379633958785336, 3.10659549353048020388584107784, 4.10939583642853321779679211717, 5.25119821181268042392866551484, 5.84338736633590291479134862546, 6.69513488283933821665759503446, 7.33429523527432624177834362891, 8.387772240300479727383083834653, 9.058661210836890141999244815746

Graph of the ZZ-function along the critical line