Properties

Degree $2$
Conductor $1827$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 2·5-s + 7-s − 3·8-s + 2·10-s − 4·11-s − 2·13-s + 14-s − 16-s − 2·17-s − 4·19-s − 2·20-s − 4·22-s − 25-s − 2·26-s − 28-s − 29-s − 8·31-s + 5·32-s − 2·34-s + 2·35-s − 10·37-s − 4·38-s − 6·40-s + 6·41-s + 12·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 1.06·8-s + 0.632·10-s − 1.20·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.185·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s + 0.338·35-s − 1.64·37-s − 0.648·38-s − 0.948·40-s + 0.937·41-s + 1.82·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1827 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1827 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1827\)    =    \(3^{2} \cdot 7 \cdot 29\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1827} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1827,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 - T \)
29 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−19.09457364985794, −18.55205674281973, −17.79003902993652, −17.50877141393781, −16.76748215707568, −15.67693002798884, −15.23335436062556, −14.37321750368580, −13.92134120716589, −13.30969301681599, −12.60673543804569, −12.24360153444879, −10.86390088649027, −10.59211353740616, −9.436880328734883, −9.082806523501985, −8.078177291112253, −7.268774935367258, −6.092470603331158, −5.548265673919248, −4.837572681315123, −4.029061328137380, −2.802880002148125, −1.964099516657620, 0, 1.964099516657620, 2.802880002148125, 4.029061328137380, 4.837572681315123, 5.548265673919248, 6.092470603331158, 7.268774935367258, 8.078177291112253, 9.082806523501985, 9.436880328734883, 10.59211353740616, 10.86390088649027, 12.24360153444879, 12.60673543804569, 13.30969301681599, 13.92134120716589, 14.37321750368580, 15.23335436062556, 15.67693002798884, 16.76748215707568, 17.50877141393781, 17.79003902993652, 18.55205674281973, 19.09457364985794

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