↑

Properties

Label 2-3864-3864.3443-c0-0-2
Degree $2$
Conductor $3864$
Sign $-0.763 + 0.645i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.142 + 0.989i)6-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)12-s + (1.25 + 0.368i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (0.118 − 0.822i)17-s + (−0.415 − 0.909i)18-s + (0.654 − 0.755i)21-s + (−0.841 − 0.540i)23-s + 24-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.142 + 0.989i)6-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)12-s + (1.25 + 0.368i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (0.118 − 0.822i)17-s + (−0.415 − 0.909i)18-s + (0.654 − 0.755i)21-s + (−0.841 − 0.540i)23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.763 + 0.645i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (3443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ -0.763 + 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8591478630\)
\(L(\frac12)\) \(\approx\) \(0.8591478630\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 + 0.755i)T \)
3 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
good5 \( 1 + (0.654 - 0.755i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
17 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (-0.239 + 1.66i)T + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.345 + 0.755i)T + (-0.654 + 0.755i)T^{2} \)
43 \( 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (1.84 + 0.540i)T + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.841 - 0.540i)T^{2} \)
83 \( 1 + (-0.544 + 1.19i)T + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (0.654 - 0.755i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.753338010512090272808855038271, −7.44444333845762263518282662034, −6.35398406906187232495443411840, −6.10666302286452707759275771098, −5.40349973912086251055010059866, −4.39588896237019966494864209646, −3.82382593700787759418488318223, −3.07214838311637436780246557086, −1.88288112473790507444242735772, −0.44653285970787922910637549761, 1.40837933421023581832410634056, 2.87320333135196834436784998666, 3.74861213559815620046291095312, 4.46016333571088803067149757110, 5.53132564093673079145427995876, 6.14555789757932168439638769898, 6.40187033576213476651221679735, 7.35790894346574237011523697543, 7.920425493633805025050884919222, 8.682232295573262543856579856552

Graph of the $Z$-function along the critical line