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Properties

Label 2-3864-3864.2603-c0-0-3
Degree $2$
Conductor $3864$
Sign $0.117 + 0.993i$
Analytic cond. $1.92838$
Root an. cond. $1.38866$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.117 + 0.993i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (2603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ 0.117 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.476205546\)
\(L(\frac12)\) \(\approx\) \(1.476205546\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (-0.959 + 0.281i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
good5 \( 1 + (-0.415 - 0.909i)T^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
17 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (1.25 + 1.45i)T + (-0.142 + 0.989i)T^{2} \)
31 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.415 + 0.909i)T^{2} \)
41 \( 1 + (1.41 + 0.909i)T + (0.415 + 0.909i)T^{2} \)
43 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.0405 - 0.281i)T + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
67 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.118 - 0.258i)T + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.959 + 0.281i)T^{2} \)
83 \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \)
89 \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.415 - 0.909i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line