Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 0.759·2-s − 1.42·4-s − 2.86·5-s + 7-s − 2.59·8-s − 2.17·10-s − 6.96·13-s + 0.759·14-s + 0.872·16-s − 6.65·17-s − 5.75·19-s + 4.07·20-s + 0.724·23-s + 3.20·25-s − 5.28·26-s − 1.42·28-s − 7.56·29-s + 2.81·31-s + 5.86·32-s − 5.05·34-s − 2.86·35-s − 1.72·37-s − 4.36·38-s + 7.44·40-s − 1.12·41-s − 11.7·43-s + 0.550·46-s + ⋯
L(s)  = 1  + 0.536·2-s − 0.711·4-s − 1.28·5-s + 0.377·7-s − 0.919·8-s − 0.688·10-s − 1.93·13-s + 0.202·14-s + 0.218·16-s − 1.61·17-s − 1.31·19-s + 0.911·20-s + 0.151·23-s + 0.641·25-s − 1.03·26-s − 0.268·28-s − 1.40·29-s + 0.506·31-s + 1.03·32-s − 0.867·34-s − 0.484·35-s − 0.283·37-s − 0.708·38-s + 1.17·40-s − 0.175·41-s − 1.78·43-s + 0.0811·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.06326595504$
$L(\frac12)$  $\approx$  $0.06326595504$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - 0.759T + 2T^{2} \)
5 \( 1 + 2.86T + 5T^{2} \)
13 \( 1 + 6.96T + 13T^{2} \)
17 \( 1 + 6.65T + 17T^{2} \)
19 \( 1 + 5.75T + 19T^{2} \)
23 \( 1 - 0.724T + 23T^{2} \)
29 \( 1 + 7.56T + 29T^{2} \)
31 \( 1 - 2.81T + 31T^{2} \)
37 \( 1 + 1.72T + 37T^{2} \)
41 \( 1 + 1.12T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + 5.18T + 47T^{2} \)
53 \( 1 - 6.25T + 53T^{2} \)
59 \( 1 + 3.98T + 59T^{2} \)
61 \( 1 + 5.97T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 0.995T + 71T^{2} \)
73 \( 1 - 3.53T + 73T^{2} \)
79 \( 1 + 0.669T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.025216635145892040904589538661, −7.09047989311428657974118981792, −6.65287043682681844032329974927, −5.50847924541665058424284092970, −4.79500245251247721203925643261, −4.39710977392830519811786780289, −3.81759411658507109144113435365, −2.84581881764334530959224184263, −1.97148598847601426630480731709, −0.10880616824171949175985256250, 0.10880616824171949175985256250, 1.97148598847601426630480731709, 2.84581881764334530959224184263, 3.81759411658507109144113435365, 4.39710977392830519811786780289, 4.79500245251247721203925643261, 5.50847924541665058424284092970, 6.65287043682681844032329974927, 7.09047989311428657974118981792, 8.025216635145892040904589538661

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