Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 9-s + 6·11-s − 3·13-s − 2·15-s − 4·17-s − 5·19-s − 4·23-s − 25-s − 27-s + 4·29-s − 7·31-s − 6·33-s + 9·37-s + 3·39-s + 2·41-s + 43-s + 2·45-s − 2·47-s + 4·51-s − 8·53-s + 12·55-s + 5·57-s + 10·61-s − 6·65-s + 15·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.80·11-s − 0.832·13-s − 0.516·15-s − 0.970·17-s − 1.14·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.742·29-s − 1.25·31-s − 1.04·33-s + 1.47·37-s + 0.480·39-s + 0.312·41-s + 0.152·43-s + 0.298·45-s − 0.291·47-s + 0.560·51-s − 1.09·53-s + 1.61·55-s + 0.662·57-s + 1.28·61-s − 0.744·65-s + 1.83·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{9408} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.878115463\)
\(L(\frac12)\)  \(\approx\)  \(1.878115463\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−7.57253296889655998207872955026, −6.72811271187868704365785784722, −6.34228447368293208534813513816, −5.87834938662477737684078141377, −4.89256338817006386774221993848, −4.30483386019872243022458773634, −3.63005399864809429529145470299, −2.24607079808979957145818625772, −1.87850979913662677038990110622, −0.67051282481112097276120489509, 0.67051282481112097276120489509, 1.87850979913662677038990110622, 2.24607079808979957145818625772, 3.63005399864809429529145470299, 4.30483386019872243022458773634, 4.89256338817006386774221993848, 5.87834938662477737684078141377, 6.34228447368293208534813513816, 6.72811271187868704365785784722, 7.57253296889655998207872955026

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