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  + 2·2-s + 2·4-s + 3·5-s + 7-s + 6·10-s − 2·13-s + 2·14-s − 4·16-s + 3·17-s + 4·19-s + 6·20-s − 2·23-s + 4·25-s − 4·26-s + 2·28-s + 8·29-s + 2·31-s − 8·32-s + 6·34-s + 3·35-s + 10·37-s + 8·38-s + 2·41-s − 9·43-s − 4·46-s − 9·47-s + 49-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.34·5-s + 0.377·7-s + 1.89·10-s − 0.554·13-s + 0.534·14-s − 16-s + 0.727·17-s + 0.917·19-s + 1.34·20-s − 0.417·23-s + 4/5·25-s − 0.784·26-s + 0.377·28-s + 1.48·29-s + 0.359·31-s − 1.41·32-s + 1.02·34-s + 0.507·35-s + 1.64·37-s + 1.29·38-s + 0.312·41-s − 1.37·43-s − 0.589·46-s − 1.31·47-s + 1/7·49-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  :  $\chi_{7623} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.248007660$
$L(\frac12)$  $\approx$  $6.248007660$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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

−17.02647191174165, −16.30061916164659, −15.85481040133376, −14.80339204083534, −14.70618974206518, −13.97954413031015, −13.65629784950069, −13.07015072254998, −12.50736816698045, −11.75247107939374, −11.49580743873618, −10.37940991611716, −9.873722764972044, −9.424326801419145, −8.466482339605520, −7.762114776897596, −6.767007832158670, −6.321491722416356, −5.538717065785345, −5.160667561455604, −4.516559198319617, −3.572941808948763, −2.745396492661509, −2.169983523245532, −1.057241578281989, 1.057241578281989, 2.169983523245532, 2.745396492661509, 3.572941808948763, 4.516559198319617, 5.160667561455604, 5.538717065785345, 6.321491722416356, 6.767007832158670, 7.762114776897596, 8.466482339605520, 9.424326801419145, 9.873722764972044, 10.37940991611716, 11.49580743873618, 11.75247107939374, 12.50736816698045, 13.07015072254998, 13.65629784950069, 13.97954413031015, 14.70618974206518, 14.80339204083534, 15.85481040133376, 16.30061916164659, 17.02647191174165

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