Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 5·4-s + 2·7-s − 20·8-s − 3·9-s + 20·11-s + 4·14-s + 5·16-s − 6·18-s + 40·22-s − 28·23-s + 2·25-s − 10·28-s − 76·29-s + 118·32-s + 15·36-s + 52·37-s + 52·43-s − 100·44-s − 56·46-s − 45·49-s + 4·50-s + 20·53-s − 40·56-s − 152·58-s − 6·63-s + 111·64-s + ⋯
L(s)  = 1  + 2-s − 5/4·4-s + 2/7·7-s − 5/2·8-s − 1/3·9-s + 1.81·11-s + 2/7·14-s + 5/16·16-s − 1/3·18-s + 1.81·22-s − 1.21·23-s + 2/25·25-s − 0.357·28-s − 2.62·29-s + 3.68·32-s + 5/12·36-s + 1.40·37-s + 1.20·43-s − 2.27·44-s − 1.21·46-s − 0.918·49-s + 2/25·50-s + 0.377·53-s − 5/7·56-s − 2.62·58-s − 0.0952·63-s + 1.73·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(441\)    =    \(3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{21} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 441,\ (\ :1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.912447$
$L(\frac12)$  $\approx$  $0.912447$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 - 2 T + p^{2} T^{2} \)
good2$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{4} T^{4} \)
11$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
19$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
23$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 38 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1154 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 1438 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3650 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 1154 T^{2} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 6242 T^{2} + p^{4} T^{4} \)
67$C_2$ \( ( 1 - 74 T + p^{2} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 62 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 8930 T^{2} + p^{4} T^{4} \)
79$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 5666 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 14114 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 15746 T^{2} + p^{4} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.10517069117350486220117120460, −17.81695983604413400532376722810, −16.99709269874029433437998324888, −16.68516973657824329419909365056, −15.41901003723195167156855700492, −14.75742959313816807996059400352, −14.19899250884021282252848878322, −14.14940590373736655263465816724, −13.01727907733575660680791099605, −12.77709226594676102208899896857, −11.76131059760542062142366906094, −11.40828011099156541309646669122, −9.904220807212029954536797316677, −9.278044820705301326040810468881, −8.795882945302853429926271393819, −7.76342273456707659248606363139, −6.22659888357279980443063344171, −5.56031607780747393425907988408, −4.31814990297491096689755315369, −3.71728929475028991669282655984, 3.71728929475028991669282655984, 4.31814990297491096689755315369, 5.56031607780747393425907988408, 6.22659888357279980443063344171, 7.76342273456707659248606363139, 8.795882945302853429926271393819, 9.278044820705301326040810468881, 9.904220807212029954536797316677, 11.40828011099156541309646669122, 11.76131059760542062142366906094, 12.77709226594676102208899896857, 13.01727907733575660680791099605, 14.14940590373736655263465816724, 14.19899250884021282252848878322, 14.75742959313816807996059400352, 15.41901003723195167156855700492, 16.68516973657824329419909365056, 16.99709269874029433437998324888, 17.81695983604413400532376722810, 18.10517069117350486220117120460

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