Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{3} $
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  − 3·4-s + 5-s + 9-s − 8·11-s + 5·16-s + 8·19-s − 3·20-s + 25-s − 4·29-s − 3·36-s + 20·41-s + 24·44-s + 45-s − 14·49-s − 8·55-s − 8·59-s − 4·61-s − 3·64-s − 16·71-s − 24·76-s + 5·80-s + 81-s − 12·89-s + 8·95-s − 8·99-s − 3·100-s + 12·101-s + ⋯
L(s)  = 1  − 3/2·4-s + 0.447·5-s + 1/3·9-s − 2.41·11-s + 5/4·16-s + 1.83·19-s − 0.670·20-s + 1/5·25-s − 0.742·29-s − 1/2·36-s + 3.12·41-s + 3.61·44-s + 0.149·45-s − 2·49-s − 1.07·55-s − 1.04·59-s − 0.512·61-s − 3/8·64-s − 1.89·71-s − 2.75·76-s + 0.559·80-s + 1/9·81-s − 1.27·89-s + 0.820·95-s − 0.804·99-s − 0.299·100-s + 1.19·101-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1125\)    =    \(3^{2} \cdot 5^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1125} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1125,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4386469691$
$L(\frac12)$  $\approx$  $0.4386469691$
$L(\frac{3}{2})$   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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−14.00692585049802430183598730465, −13.37224542302974911325065415564, −12.89741011334273074809178538613, −12.64617876135106218873747419153, −11.42796116649064334018778191223, −10.67892245123374144028858824682, −9.965958896202437987484594622667, −9.523451675812284265792380668213, −8.843956595114680827452383410006, −7.68003844385765985083943135134, −7.66488013441745380243230842523, −5.86497653842965752491328535183, −5.23920392624592057055772361749, −4.52160444884874669934496061646, −3.01549784140686353642765162463, 3.01549784140686353642765162463, 4.52160444884874669934496061646, 5.23920392624592057055772361749, 5.86497653842965752491328535183, 7.66488013441745380243230842523, 7.68003844385765985083943135134, 8.843956595114680827452383410006, 9.523451675812284265792380668213, 9.965958896202437987484594622667, 10.67892245123374144028858824682, 11.42796116649064334018778191223, 12.64617876135106218873747419153, 12.89741011334273074809178538613, 13.37224542302974911325065415564, 14.00692585049802430183598730465

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