↑

Properties

Label 2-3864-3864.1805-c0-0-6
Degree $2$
Conductor $3864$
Sign $-0.420 + 0.907i$
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  + (−0.540 − 0.841i)2-s + (0.707 − 0.707i)3-s + (−0.415 + 0.909i)4-s + (1.91 − 0.562i)5-s + (−0.977 − 0.212i)6-s + (−0.755 + 0.654i)7-s + (0.989 − 0.142i)8-s − 1.00i·9-s + (−1.50 − 1.30i)10-s + (0.349 + 0.936i)12-s + (−0.278 + 0.321i)13-s + (0.959 + 0.281i)14-s + (0.956 − 1.75i)15-s + (−0.654 − 0.755i)16-s + (−0.841 + 0.540i)18-s + (−0.871 − 0.398i)19-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)2-s + (0.707 − 0.707i)3-s + (−0.415 + 0.909i)4-s + (1.91 − 0.562i)5-s + (−0.977 − 0.212i)6-s + (−0.755 + 0.654i)7-s + (0.989 − 0.142i)8-s − 1.00i·9-s + (−1.50 − 1.30i)10-s + (0.349 + 0.936i)12-s + (−0.278 + 0.321i)13-s + (0.959 + 0.281i)14-s + (0.956 − 1.75i)15-s + (−0.654 − 0.755i)16-s + (−0.841 + 0.540i)18-s + (−0.871 − 0.398i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.420 + 0.907i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (1805, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ -0.420 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.547802298\)
\(L(\frac12)\) \(\approx\) \(1.547802298\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.540 + 0.841i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (0.755 - 0.654i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
good5 \( 1 + (-1.91 + 0.562i)T + (0.841 - 0.540i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.278 - 0.321i)T + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.871 + 0.398i)T + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 + (0.841 + 0.540i)T^{2} \)
41 \( 1 + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (1.32 + 1.14i)T + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (-1.39 + 0.201i)T + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.909 - 1.41i)T + (-0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (0.627 + 0.544i)T + (0.142 + 0.989i)T^{2} \)
83 \( 1 + (-1.79 - 0.527i)T + (0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.841 - 0.540i)T^{2} \)
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

−8.821909880058964407590623421959, −8.048907810174124362626690511268, −6.74888233435351057078185728266, −6.51952637045638658912421679885, −5.46999313127439177149571003756, −4.57831380701445038584586468137, −3.32081032384532053714761543635, −2.33165367268574661940092678457, −2.20364534909946057908149406721, −1.00653569122493925622874012889, 1.52983133232425032106895364990, 2.47632888704576036621912572059, 3.39685752099703061172678966276, 4.53101611670198899459677098294, 5.39534457030124459911991299071, 5.98163567304708391145265674894, 6.74249705551816288855145573541, 7.32863689279373450907660193772, 8.255523750147351069847991209314, 9.169430813776084162407891953355

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