Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \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  − 6·11-s − 2·13-s − 4·19-s − 6·23-s − 5·25-s − 6·29-s + 8·31-s + 2·37-s + 12·41-s + 4·43-s − 12·47-s + 6·53-s + 10·61-s − 8·67-s + 6·71-s + 10·73-s + 4·79-s + 12·83-s + 12·89-s + 10·97-s − 12·101-s + 8·103-s − 6·107-s + 14·109-s + 6·113-s + ⋯
L(s)  = 1  − 1.80·11-s − 0.554·13-s − 0.917·19-s − 1.25·23-s − 25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s + 1.87·41-s + 0.609·43-s − 1.75·47-s + 0.824·53-s + 1.28·61-s − 0.977·67-s + 0.712·71-s + 1.17·73-s + 0.450·79-s + 1.31·83-s + 1.27·89-s + 1.01·97-s − 1.19·101-s + 0.788·103-s − 0.580·107-s + 1.34·109-s + 0.564·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7056} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7056,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.043028347\)
\(L(\frac12)\)  \(\approx\)  \(1.043028347\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−7.84146975197230387968728229721, −7.53496789087848454636436751023, −6.41711207711527362534874367136, −5.85790282414911520929854670202, −5.11307774197202177256002368382, −4.41403780256165179403663455868, −3.59967552507315280398412985359, −2.46425622819866198981434285051, −2.13744105481676125327589472523, −0.48483142066886114300580830013, 0.48483142066886114300580830013, 2.13744105481676125327589472523, 2.46425622819866198981434285051, 3.59967552507315280398412985359, 4.41403780256165179403663455868, 5.11307774197202177256002368382, 5.85790282414911520929854670202, 6.41711207711527362534874367136, 7.53496789087848454636436751023, 7.84146975197230387968728229721

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