Properties

Degree 4
Conductor $ 3 \cdot 5^{2} \cdot 17^{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  − 3-s − 4-s − 2·7-s + 12-s − 3·16-s − 6·17-s + 4·19-s + 2·21-s − 6·23-s − 5·25-s + 4·27-s + 2·28-s − 8·37-s + 3·48-s − 2·49-s + 6·51-s − 4·57-s − 12·59-s + 7·64-s + 6·68-s + 6·69-s + 16·73-s + 5·75-s − 4·76-s − 7·81-s − 2·84-s + 6·92-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 0.755·7-s + 0.288·12-s − 3/4·16-s − 1.45·17-s + 0.917·19-s + 0.436·21-s − 1.25·23-s − 25-s + 0.769·27-s + 0.377·28-s − 1.31·37-s + 0.433·48-s − 2/7·49-s + 0.840·51-s − 0.529·57-s − 1.56·59-s + 7/8·64-s + 0.727·68-s + 0.722·69-s + 1.87·73-s + 0.577·75-s − 0.458·76-s − 7/9·81-s − 0.218·84-s + 0.625·92-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(21675\)    =    \(3 \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{21675} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 21675,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( 1 + p T^{2} \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.64633384544385515680215023732, −9.869886958627660833094493302004, −9.518419820493640327168381820105, −9.066172426018826794796087207332, −8.372850595683571950500622849438, −7.88521058464608532391749024144, −6.89517010030384008110494764675, −6.71103886476802877027475809592, −5.94515789337932256796297517782, −5.36848475646101974564338342579, −4.59172082351411232785791696242, −4.03324695467829362811994557437, −3.13165478752451171735177074911, −2.02226205952559193812136055481, 0, 2.02226205952559193812136055481, 3.13165478752451171735177074911, 4.03324695467829362811994557437, 4.59172082351411232785791696242, 5.36848475646101974564338342579, 5.94515789337932256796297517782, 6.71103886476802877027475809592, 6.89517010030384008110494764675, 7.88521058464608532391749024144, 8.372850595683571950500622849438, 9.066172426018826794796087207332, 9.518419820493640327168381820105, 9.869886958627660833094493302004, 10.64633384544385515680215023732

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