Properties

Degree 4
Conductor $ 3^{3} \cdot 5^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·4-s + 9-s + 3·12-s − 4·13-s + 5·16-s + 8·19-s + 25-s − 27-s − 3·36-s − 20·37-s + 4·39-s + 8·43-s − 5·48-s − 14·49-s + 12·52-s − 8·57-s − 4·61-s − 3·64-s + 24·67-s + 20·73-s − 75-s − 24·76-s + 81-s + 4·97-s − 3·100-s − 32·103-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 1/3·9-s + 0.866·12-s − 1.10·13-s + 5/4·16-s + 1.83·19-s + 1/5·25-s − 0.192·27-s − 1/2·36-s − 3.28·37-s + 0.640·39-s + 1.21·43-s − 0.721·48-s − 2·49-s + 1.66·52-s − 1.05·57-s − 0.512·61-s − 3/8·64-s + 2.93·67-s + 2.34·73-s − 0.115·75-s − 2.75·76-s + 1/9·81-s + 0.406·97-s − 0.299·100-s − 3.15·103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(675\)    =    \(3^{3} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{675} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 675,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3226957464$
$L(\frac12)$  $\approx$  $0.3226957464$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.55874952579683322699316330788, −14.00692585049802430183598730465, −13.71457429111566649845275587943, −12.64617876135106218873747419153, −12.40658486355792546821392015265, −11.57318950916085697381955699128, −10.67892245123374144028858824682, −9.738851148469479131038939478590, −9.523451675812284265792380668213, −8.554588868470699894187777077269, −7.66488013441745380243230842523, −6.76955885158912340897756950945, −5.23920392624592057055772361749, −5.04860090093668311608273372623, −3.63412818236731287070311550143, 3.63412818236731287070311550143, 5.04860090093668311608273372623, 5.23920392624592057055772361749, 6.76955885158912340897756950945, 7.66488013441745380243230842523, 8.554588868470699894187777077269, 9.523451675812284265792380668213, 9.738851148469479131038939478590, 10.67892245123374144028858824682, 11.57318950916085697381955699128, 12.40658486355792546821392015265, 12.64617876135106218873747419153, 13.71457429111566649845275587943, 14.00692585049802430183598730465, 14.55874952579683322699316330788

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