Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 2·17-s − 2·19-s + 8·23-s + 11·25-s − 2·29-s + 4·31-s − 6·37-s − 2·41-s − 8·43-s + 4·47-s + 10·53-s − 6·59-s − 4·61-s + 12·67-s + 14·73-s + 8·79-s − 6·83-s + 8·85-s + 10·89-s + 8·95-s + 2·97-s + 12·101-s − 12·103-s − 12·107-s + 10·109-s − 6·113-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.485·17-s − 0.458·19-s + 1.66·23-s + 11/5·25-s − 0.371·29-s + 0.718·31-s − 0.986·37-s − 0.312·41-s − 1.21·43-s + 0.583·47-s + 1.37·53-s − 0.781·59-s − 0.512·61-s + 1.46·67-s + 1.63·73-s + 0.900·79-s − 0.658·83-s + 0.867·85-s + 1.05·89-s + 0.820·95-s + 0.203·97-s + 1.19·101-s − 1.18·103-s − 1.16·107-s + 0.957·109-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  =  \(1\)
Selberg data  =  \((2,\ 7056,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−7.58028626270520696649741407666, −6.96697866536475375893625237684, −6.48033933151697843502675981340, −5.19459451107974603578328200690, −4.73247690983758697231526566319, −3.86396283353620397101485600351, −3.37079911116950031942239105021, −2.42039669908436992478027344666, −1.04001144089587954483115872500, 0, 1.04001144089587954483115872500, 2.42039669908436992478027344666, 3.37079911116950031942239105021, 3.86396283353620397101485600351, 4.73247690983758697231526566319, 5.19459451107974603578328200690, 6.48033933151697843502675981340, 6.96697866536475375893625237684, 7.58028626270520696649741407666

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