↑

Properties

Label 2-3864-3864.125-c0-0-2
Degree $2$
Conductor $3864$
Sign $0.325 - 0.945i$
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  + (−0.755 + 0.654i)2-s + (0.707 − 0.707i)3-s + (0.142 − 0.989i)4-s + (−0.729 + 1.59i)5-s + (−0.0713 + 0.997i)6-s + (−0.281 + 0.959i)7-s + (0.540 + 0.841i)8-s − 1.00i·9-s + (−0.494 − 1.68i)10-s + (−0.599 − 0.800i)12-s + (1.91 − 0.562i)13-s + (−0.415 − 0.909i)14-s + (0.613 + 1.64i)15-s + (−0.959 − 0.281i)16-s + (0.654 + 0.755i)18-s + (0.691 + 0.0994i)19-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)2-s + (0.707 − 0.707i)3-s + (0.142 − 0.989i)4-s + (−0.729 + 1.59i)5-s + (−0.0713 + 0.997i)6-s + (−0.281 + 0.959i)7-s + (0.540 + 0.841i)8-s − 1.00i·9-s + (−0.494 − 1.68i)10-s + (−0.599 − 0.800i)12-s + (1.91 − 0.562i)13-s + (−0.415 − 0.909i)14-s + (0.613 + 1.64i)15-s + (−0.959 − 0.281i)16-s + (0.654 + 0.755i)18-s + (0.691 + 0.0994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ 0.325 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063331267\)
\(L(\frac12)\) \(\approx\) \(1.063331267\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.755 - 0.654i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (0.281 - 0.959i)T \)
23 \( 1 + (-0.959 + 0.281i)T \)
good5 \( 1 + (0.729 - 1.59i)T + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (-1.91 + 0.562i)T + (0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (-0.691 - 0.0994i)T + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (-0.415 + 0.909i)T^{2} \)
37 \( 1 + (-0.654 + 0.755i)T^{2} \)
41 \( 1 + (-0.654 - 0.755i)T^{2} \)
43 \( 1 + (0.415 + 0.909i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.527 - 1.79i)T + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.764 - 1.18i)T + (-0.415 + 0.909i)T^{2} \)
67 \( 1 + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (0.989 - 0.857i)T + (0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.0801 - 0.273i)T + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.665 - 1.45i)T + (-0.654 + 0.755i)T^{2} \)
89 \( 1 + (-0.415 - 0.909i)T^{2} \)
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

−8.559894019348456703241786614791, −8.131680235477783488126985256046, −7.29662815509123612043502226223, −6.82522627264108946366304744444, −6.13461864280989785900533629938, −5.59440435344948523781351796603, −3.93767779753633820086109061461, −3.09379343540250470182153509782, −2.51317598655010363971402702464, −1.18512347596909978399004368816, 0.890431864540458737609533230502, 1.70600558628390303536954715121, 3.36216211398992226025699885641, 3.68221128545968511634189451459, 4.42357237153044756064195765846, 5.11729784007815075888221702349, 6.51334136838273299987697701294, 7.56827537349447428761459558437, 8.006075979094586508914777545584, 8.792076938573322869586056108202

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