Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 19^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 2·5-s + 9-s + 5·16-s − 2·19-s + 6·20-s − 25-s + 4·29-s + 16·31-s − 3·36-s − 4·41-s − 2·45-s − 14·49-s − 24·59-s − 4·61-s − 3·64-s + 6·76-s − 10·80-s + 81-s − 4·89-s + 4·95-s + 3·100-s − 20·101-s − 20·109-s − 12·116-s − 22·121-s − 48·124-s + ⋯
L(s)  = 1  − 3/2·4-s − 0.894·5-s + 1/3·9-s + 5/4·16-s − 0.458·19-s + 1.34·20-s − 1/5·25-s + 0.742·29-s + 2.87·31-s − 1/2·36-s − 0.624·41-s − 0.298·45-s − 2·49-s − 3.12·59-s − 0.512·61-s − 3/8·64-s + 0.688·76-s − 1.11·80-s + 1/9·81-s − 0.423·89-s + 0.410·95-s + 3/10·100-s − 1.99·101-s − 1.91·109-s − 1.11·116-s − 2·121-s − 4.31·124-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(81225\)    =    \(3^{2} \cdot 5^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{81225} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 81225,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.438835779453266646722697155473, −9.032116632714081125535639739797, −8.263513432862467471567669978548, −8.129458035743720070592981027873, −7.81423371583594124822281317820, −6.83493330235856618687394516740, −6.46787589567348650226556085189, −5.78421312004364849132957140059, −4.89453559736176945278761858504, −4.56991321080597356703354311172, −4.24814179905599513569109101908, −3.44365518362789223021999993571, −2.80168364658006648054788403582, −1.32661394670938609373916514708, 0, 1.32661394670938609373916514708, 2.80168364658006648054788403582, 3.44365518362789223021999993571, 4.24814179905599513569109101908, 4.56991321080597356703354311172, 4.89453559736176945278761858504, 5.78421312004364849132957140059, 6.46787589567348650226556085189, 6.83493330235856618687394516740, 7.81423371583594124822281317820, 8.129458035743720070592981027873, 8.263513432862467471567669978548, 9.032116632714081125535639739797, 9.438835779453266646722697155473

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