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Properties

Label 2-3864-3864.1469-c0-0-5
Degree 22
Conductor 38643864
Sign 0.0850+0.996i0.0850 + 0.996i
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  + (0.989 − 0.142i)2-s + (−0.707 − 0.707i)3-s + (0.959 − 0.281i)4-s + (1.27 − 1.47i)5-s + (−0.800 − 0.599i)6-s + (0.540 + 0.841i)7-s + (0.909 − 0.415i)8-s + 1.00i·9-s + (1.05 − 1.64i)10-s + (−0.877 − 0.479i)12-s + (−1.00 − 0.647i)13-s + (0.654 + 0.755i)14-s + (−1.94 + 0.139i)15-s + (0.841 − 0.540i)16-s + (0.142 + 0.989i)18-s + (0.562 + 1.91i)19-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)2-s + (−0.707 − 0.707i)3-s + (0.959 − 0.281i)4-s + (1.27 − 1.47i)5-s + (−0.800 − 0.599i)6-s + (0.540 + 0.841i)7-s + (0.909 − 0.415i)8-s + 1.00i·9-s + (1.05 − 1.64i)10-s + (−0.877 − 0.479i)12-s + (−1.00 − 0.647i)13-s + (0.654 + 0.755i)14-s + (−1.94 + 0.139i)15-s + (0.841 − 0.540i)16-s + (0.142 + 0.989i)18-s + (0.562 + 1.91i)19-s + ⋯

Functional equation

Λ(s)=(3864s/2 ΓC(s) L(s)=((0.0850+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0850 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3864s/2 ΓC(s) L(s)=((0.0850+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0850 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38643864    =    23⋅3⋅7⋅232^{3} \cdot 3 \cdot 7 \cdot 23
Sign: 0.0850+0.996i0.0850 + 0.996i
Analytic conductor: 1.928381.92838
Root analytic conductor: 1.388661.38866
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3864(1469,⋅)\chi_{3864} (1469, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3864, ( :0), 0.0850+0.996i)(2,\ 3864,\ (\ :0),\ 0.0850 + 0.996i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.5554756272.555475627
L(12)L(\frac12) ≈\approx 2.5554756272.555475627
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+(−0.989+0.142i)T 1 + (-0.989 + 0.142i)T
3 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
7 1+(−0.540−0.841i)T 1 + (-0.540 - 0.841i)T
23 1+(0.841+0.540i)T 1 + (0.841 + 0.540i)T
good5 1+(−1.27+1.47i)T+(−0.142−0.989i)T2 1 + (-1.27 + 1.47i)T + (-0.142 - 0.989i)T^{2}
11 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
13 1+(1.00+0.647i)T+(0.415+0.909i)T2 1 + (1.00 + 0.647i)T + (0.415 + 0.909i)T^{2}
17 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
19 1+(−0.562−1.91i)T+(−0.841+0.540i)T2 1 + (-0.562 - 1.91i)T + (-0.841 + 0.540i)T^{2}
29 1+(0.841+0.540i)T2 1 + (0.841 + 0.540i)T^{2}
31 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
37 1+(−0.142+0.989i)T2 1 + (-0.142 + 0.989i)T^{2}
41 1+(−0.142−0.989i)T2 1 + (-0.142 - 0.989i)T^{2}
43 1+(−0.654−0.755i)T2 1 + (-0.654 - 0.755i)T^{2}
47 1+T2 1 + T^{2}
53 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
59 1+(−0.0771+0.120i)T+(−0.415−0.909i)T2 1 + (-0.0771 + 0.120i)T + (-0.415 - 0.909i)T^{2}
61 1+(1.28−0.587i)T+(0.654−0.755i)T2 1 + (1.28 - 0.587i)T + (0.654 - 0.755i)T^{2}
67 1+(−0.959+0.281i)T2 1 + (-0.959 + 0.281i)T^{2}
71 1+(−0.281+0.0405i)T+(0.959−0.281i)T2 1 + (-0.281 + 0.0405i)T + (0.959 - 0.281i)T^{2}
73 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
79 1+(1.03−1.61i)T+(−0.415−0.909i)T2 1 + (1.03 - 1.61i)T + (-0.415 - 0.909i)T^{2}
83 1+(−0.627−0.724i)T+(−0.142+0.989i)T2 1 + (-0.627 - 0.724i)T + (-0.142 + 0.989i)T^{2}
89 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
97 1+(−0.142−0.989i)T2 1 + (-0.142 - 0.989i)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

−8.113324756003783343132217925604, −7.922420131153898403762010316069, −6.64965658658003376855375532151, −5.75250068934605507006546602818, −5.63954859137009577936624330743, −5.00482196527189118347388813982, −4.26420716543407216676938122673, −2.63390043299699377778847581139, −1.91948979172214567275004708664, −1.25874774407091825794388883623, 1.73990119060196406722651468546, 2.69815964516065332966589319479, 3.45825402182195360685782963281, 4.48074680769946711278432582614, 5.04601441772977697787159186831, 5.81946424369593520707779337466, 6.54063832210163758630720337954, 7.07423750435451200259485731331, 7.55089710508993660993961169283, 9.138527086834462205814548551171

Graph of the ZZ-function along the critical line