Properties

Label 2-3864-1.1-c1-0-48
Degree $2$
Conductor $3864$
Sign $-1$
Analytic cond. $30.8541$
Root an. cond. $5.55465$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.27·5-s − 7-s + 9-s − 1.42·11-s + 4.69·13-s − 3.27·15-s + 3.96·17-s − 5.42·19-s − 21-s + 23-s + 5.69·25-s + 27-s + 3.96·29-s − 8.81·31-s − 1.42·33-s + 3.27·35-s + 4.81·37-s + 4.69·39-s + 2.57·41-s − 11.5·43-s − 3.27·45-s + 5.39·47-s + 49-s + 3.96·51-s − 12.6·53-s + 4.66·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.46·5-s − 0.377·7-s + 0.333·9-s − 0.429·11-s + 1.30·13-s − 0.844·15-s + 0.962·17-s − 1.24·19-s − 0.218·21-s + 0.208·23-s + 1.13·25-s + 0.192·27-s + 0.736·29-s − 1.58·31-s − 0.248·33-s + 0.552·35-s + 0.792·37-s + 0.752·39-s + 0.402·41-s − 1.76·43-s − 0.487·45-s + 0.786·47-s + 0.142·49-s + 0.555·51-s − 1.73·53-s + 0.628·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(30.8541\)
Root analytic conductor: \(5.55465\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3864,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 - T \)
good5 \( 1 + 3.27T + 5T^{2} \)
11 \( 1 + 1.42T + 11T^{2} \)
13 \( 1 - 4.69T + 13T^{2} \)
17 \( 1 - 3.96T + 17T^{2} \)
19 \( 1 + 5.42T + 19T^{2} \)
29 \( 1 - 3.96T + 29T^{2} \)
31 \( 1 + 8.81T + 31T^{2} \)
37 \( 1 - 4.81T + 37T^{2} \)
41 \( 1 - 2.57T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 - 5.39T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 - 5.51T + 67T^{2} \)
71 \( 1 + 2.69T + 71T^{2} \)
73 \( 1 + 9.35T + 73T^{2} \)
79 \( 1 + 4.57T + 79T^{2} \)
83 \( 1 + 0.277T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 7.96T + 97T^{2} \)
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

−8.099009569836202523171985594225, −7.61240301333667005159937732733, −6.76284324489530913474083377115, −6.00285926038154593434048896675, −4.92611070714413072711904277125, −3.98616931194630644003386509535, −3.56774557547826459412836660006, −2.74261835464045197543178945395, −1.36571280204201067351141203407, 0, 1.36571280204201067351141203407, 2.74261835464045197543178945395, 3.56774557547826459412836660006, 3.98616931194630644003386509535, 4.92611070714413072711904277125, 6.00285926038154593434048896675, 6.76284324489530913474083377115, 7.61240301333667005159937732733, 8.099009569836202523171985594225

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