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Properties

Label 2-3864-3864.2603-c0-0-3
Degree 22
Conductor 38643864
Sign 0.117+0.993i0.117 + 0.993i
Analytic cond. 1.928381.92838
Root an. cond. 1.388661.38866
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  + (−0.415 − 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.654 − 0.755i)6-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)12-s + (−0.118 + 0.822i)13-s + (−0.841 + 0.540i)14-s + (−0.142 − 0.989i)16-s + (1.10 + 1.27i)17-s + (−0.841 − 0.540i)18-s + (−0.415 − 0.909i)21-s + (0.959 + 0.281i)23-s + 0.999·24-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.654 − 0.755i)6-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)12-s + (−0.118 + 0.822i)13-s + (−0.841 + 0.540i)14-s + (−0.142 − 0.989i)16-s + (1.10 + 1.27i)17-s + (−0.841 − 0.540i)18-s + (−0.415 − 0.909i)21-s + (0.959 + 0.281i)23-s + 0.999·24-s + ⋯

Functional equation

Λ(s)=(3864s/2 ΓC(s) L(s)=((0.117+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3864s/2 ΓC(s) L(s)=((0.117+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38643864    =    23⋅3⋅7⋅232^{3} \cdot 3 \cdot 7 \cdot 23
Sign: 0.117+0.993i0.117 + 0.993i
Analytic conductor: 1.928381.92838
Root analytic conductor: 1.388661.38866
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3864(2603,⋅)\chi_{3864} (2603, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3864, ( :0), 0.117+0.993i)(2,\ 3864,\ (\ :0),\ 0.117 + 0.993i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4762055461.476205546
L(12)L(\frac12) ≈\approx 1.4762055461.476205546
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+(0.415+0.909i)T 1 + (0.415 + 0.909i)T
3 1+(−0.959+0.281i)T 1 + (-0.959 + 0.281i)T
7 1+(0.142+0.989i)T 1 + (0.142 + 0.989i)T
23 1+(−0.959−0.281i)T 1 + (-0.959 - 0.281i)T
good5 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
11 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
13 1+(0.118−0.822i)T+(−0.959−0.281i)T2 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2}
17 1+(−1.10−1.27i)T+(−0.142+0.989i)T2 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2}
19 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
29 1+(1.25+1.45i)T+(−0.142+0.989i)T2 1 + (1.25 + 1.45i)T + (-0.142 + 0.989i)T^{2}
31 1+(−1.25−0.368i)T+(0.841+0.540i)T2 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2}
37 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
41 1+(1.41+0.909i)T+(0.415+0.909i)T2 1 + (1.41 + 0.909i)T + (0.415 + 0.909i)T^{2}
43 1+(−1.84+0.540i)T+(0.841−0.540i)T2 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2}
47 1−T2 1 - T^{2}
53 1+(0.186+1.29i)T+(−0.959+0.281i)T2 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2}
59 1+(0.0405−0.281i)T+(−0.959−0.281i)T2 1 + (0.0405 - 0.281i)T + (-0.959 - 0.281i)T^{2}
61 1+(−0.273−0.0801i)T+(0.841+0.540i)T2 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2}
67 1+(0.118+0.258i)T+(−0.654+0.755i)T2 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2}
71 1+(−0.118−0.258i)T+(−0.654+0.755i)T2 1 + (-0.118 - 0.258i)T + (-0.654 + 0.755i)T^{2}
73 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
79 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
83 1+(0.698−0.449i)T+(0.415−0.909i)T2 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2}
89 1+(1.25−0.368i)T+(0.841−0.540i)T2 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2}
97 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{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

−8.554464032214440340359097991187, −7.87908959964982568768837954309, −7.30934168755651771840938097881, −6.62729866966941309295511570267, −5.30933198207126927515106945093, −4.12482177645325873928252987915, −3.79928592203321097710347647978, −2.95214911705127405713504010768, −1.87955938062060410866079736869, −1.10456139279841582313102518272, 1.21666804948121053734413287137, 2.64292601468562311014210780838, 3.21718435356352445582691380857, 4.53434581220936736908657292688, 5.14876531369274457218368026233, 5.83793453035278798785897143418, 6.84060730122563490289053965742, 7.50857847676502265957003936033, 8.138754514317786035903442239877, 8.777746554833905969510559084454

Graph of the ZZ-function along the critical line