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Properties

Label 2-2160-15.14-c2-0-2
Degree 22
Conductor 21602160
Sign −0.0999−0.994i-0.0999 - 0.994i
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  + (−0.5 − 4.97i)5-s − 9.94i·7-s − 9.94i·11-s + 19.8i·13-s − 22·17-s + 4·19-s − 20·23-s + (−24.5 + 4.97i)25-s + 39.7i·29-s − 29·31-s + (−49.5 + 4.97i)35-s − 39.7i·41-s + 19.8i·43-s + 58·47-s − 50·49-s + ⋯
L(s)  = 1  + (−0.100 − 0.994i)5-s − 1.42i·7-s − 0.904i·11-s + 1.53i·13-s − 1.29·17-s + 0.210·19-s − 0.869·23-s + (−0.979 + 0.198i)25-s + 1.37i·29-s − 0.935·31-s + (−1.41 + 0.142i)35-s − 0.970i·41-s + 0.462i·43-s + 1.23·47-s − 1.02·49-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.0999−0.994i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0999 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((−0.0999−0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0999 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.0999−0.994i-0.0999 - 0.994i
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), −0.0999−0.994i)(2,\ 2160,\ (\ :1),\ -0.0999 - 0.994i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.23138486070.2313848607
L(12)L(\frac12) ≈\approx 0.23138486070.2313848607
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+(0.5+4.97i)T 1 + (0.5 + 4.97i)T
good7 1+9.94iT−49T2 1 + 9.94iT - 49T^{2}
11 1+9.94iT−121T2 1 + 9.94iT - 121T^{2}
13 1−19.8iT−169T2 1 - 19.8iT - 169T^{2}
17 1+22T+289T2 1 + 22T + 289T^{2}
19 1−4T+361T2 1 - 4T + 361T^{2}
23 1+20T+529T2 1 + 20T + 529T^{2}
29 1−39.7iT−841T2 1 - 39.7iT - 841T^{2}
31 1+29T+961T2 1 + 29T + 961T^{2}
37 1−1.36e3T2 1 - 1.36e3T^{2}
41 1+39.7iT−1.68e3T2 1 + 39.7iT - 1.68e3T^{2}
43 1−19.8iT−1.84e3T2 1 - 19.8iT - 1.84e3T^{2}
47 1−58T+2.20e3T2 1 - 58T + 2.20e3T^{2}
53 1+31T+2.80e3T2 1 + 31T + 2.80e3T^{2}
59 1−39.7iT−3.48e3T2 1 - 39.7iT - 3.48e3T^{2}
61 1−44T+3.72e3T2 1 - 44T + 3.72e3T^{2}
67 1+19.8iT−4.48e3T2 1 + 19.8iT - 4.48e3T^{2}
71 1+59.6iT−5.04e3T2 1 + 59.6iT - 5.04e3T^{2}
73 1−89.5iT−5.32e3T2 1 - 89.5iT - 5.32e3T^{2}
79 1−10T+6.24e3T2 1 - 10T + 6.24e3T^{2}
83 1−19T+6.88e3T2 1 - 19T + 6.88e3T^{2}
89 1−59.6iT−7.92e3T2 1 - 59.6iT - 7.92e3T^{2}
97 1−129.iT−9.40e3T2 1 - 129. 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

−8.986531882751316602630882399122, −8.508643937876366161937867662059, −7.46072827074724363018109980971, −6.89082890919588987551501987711, −5.98324647216416591507142801750, −4.93585998804747950058982547326, −4.16729897308219768188476270166, −3.67142608662160561154501678571, −2.03304195073336264616727105349, −1.04621764566211910053047186919, 0.06066150446065525044829889174, 2.09921409985741896898227863919, 2.57082731443644731095264274816, 3.61172093325595762933765424036, 4.70075319463972335738021373770, 5.72040290600090215427023308506, 6.18598165294859868770615559754, 7.18843931977704982789713276515, 7.894669895955875854012691266338, 8.632636516939982820767213749588

Graph of the ZZ-function along the critical line