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Properties

Label 2-2160-240.83-c0-0-1
Degree 22
Conductor 21602160
Sign 0.584+0.811i0.584 + 0.811i
Analytic cond. 1.077981.07798
Root an. cond. 1.038251.03825
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 − 0.707i)8-s + 1.00·10-s + (0.707 − 0.707i)11-s + i·13-s − 1.00·16-s + (0.707 + 0.707i)17-s + (0.707 − 0.707i)20-s − 1.00i·22-s + (0.707 − 0.707i)23-s + 1.00i·25-s + (0.707 + 0.707i)26-s + (−0.707 − 0.707i)29-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 − 0.707i)8-s + 1.00·10-s + (0.707 − 0.707i)11-s + i·13-s − 1.00·16-s + (0.707 + 0.707i)17-s + (0.707 − 0.707i)20-s − 1.00i·22-s + (0.707 − 0.707i)23-s + 1.00i·25-s + (0.707 + 0.707i)26-s + (−0.707 − 0.707i)29-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.584+0.811i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s) L(s)=((0.584+0.811i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.584+0.811i0.584 + 0.811i
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2160(323,⋅)\chi_{2160} (323, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :0), 0.584+0.811i)(2,\ 2160,\ (\ :0),\ 0.584 + 0.811i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9609420021.960942002
L(12)L(\frac12) ≈\approx 1.9609420021.960942002
L(1)L(1) 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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
3 1 1
5 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good7 1−iT2 1 - iT^{2}
11 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
13 1−iT−T2 1 - iT - T^{2}
17 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
19 1+iT2 1 + iT^{2}
23 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
29 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
31 1+iT−T2 1 + iT - T^{2}
37 1−T2 1 - T^{2}
41 1+1.41T+T2 1 + 1.41T + T^{2}
43 1−iT−T2 1 - iT - T^{2}
47 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
53 1+1.41iT−T2 1 + 1.41iT - T^{2}
59 1+iT2 1 + iT^{2}
61 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
67 1−2iT−T2 1 - 2iT - T^{2}
71 1−T2 1 - T^{2}
73 1−iT2 1 - iT^{2}
79 1+T+T2 1 + T + T^{2}
83 1+1.41iT−T2 1 + 1.41iT - T^{2}
89 1−T2 1 - T^{2}
97 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{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

−9.354013577683039362561979008138, −8.658507239135255416879567362128, −7.37359938631955290868199707454, −6.38984767765554831714236969744, −6.11291575861926805959168187419, −5.12733775927277385797823846845, −4.06782762135547836546210517358, −3.35839383917449221237263436594, −2.34840303986454687848843743689, −1.41097822543807620976198449082, 1.49683688304972513288255829459, 2.85830229168009217376887100185, 3.75139535317117595261612570909, 4.91329552612959679352130677452, 5.28538032218939531118593246962, 6.10055728291232742287314563741, 7.06118132599108263225895833294, 7.59671826868683794236910766877, 8.646691367825217732399887287720, 9.154028413843418373892626934308

Graph of the ZZ-function along the critical line