Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s + 3·5-s − 2·9-s + 12-s + 3·15-s − 3·16-s + 3·20-s + 4·25-s − 5·27-s + 5·31-s − 2·36-s + 6·37-s − 6·45-s − 3·48-s + 4·49-s − 15·53-s + 3·60-s − 7·64-s + 9·67-s − 15·71-s + 4·75-s − 9·80-s + 81-s − 15·89-s + 5·93-s − 6·97-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 1.34·5-s − 2/3·9-s + 0.288·12-s + 0.774·15-s − 3/4·16-s + 0.670·20-s + 4/5·25-s − 0.962·27-s + 0.898·31-s − 1/3·36-s + 0.986·37-s − 0.894·45-s − 0.433·48-s + 4/7·49-s − 2.06·53-s + 0.387·60-s − 7/8·64-s + 1.09·67-s − 1.78·71-s + 0.461·75-s − 1.00·80-s + 1/9·81-s − 1.58·89-s + 0.518·93-s − 0.609·97-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(27225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{27225} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 27225,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.868536700$
$L(\frac12)$  $\approx$  $1.868536700$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 - T + p T^{2} \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good2$V_4$ \( 1 - T^{2} + p^{2} T^{4} \)
7$V_4$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$V_4$ \( 1 - T^{2} + p^{2} T^{4} \)
17$V_4$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$V_4$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$V_4$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
43$V_4$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$V_4$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$V_4$ \( 1 - 91 T^{2} + p^{2} T^{4} \)
79$V_4$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 + 101 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−10.61804269149265698780892668657, −9.882401166416226415873153865669, −9.590601188009767510762478021927, −9.088612380213389799186881556024, −8.527871540816042242216556571634, −8.003754542377893610921486397718, −7.33255041163428354031584364016, −6.63194036733281711018312364189, −6.15394990959925820699910873782, −5.68591232663550540863181236433, −4.93041013461069892960488838985, −4.14295500345574971300811236329, −2.99719913411464429328347298085, −2.55008190051617150431531328373, −1.69497144263362379198334245522, 1.69497144263362379198334245522, 2.55008190051617150431531328373, 2.99719913411464429328347298085, 4.14295500345574971300811236329, 4.93041013461069892960488838985, 5.68591232663550540863181236433, 6.15394990959925820699910873782, 6.63194036733281711018312364189, 7.33255041163428354031584364016, 8.003754542377893610921486397718, 8.527871540816042242216556571634, 9.088612380213389799186881556024, 9.590601188009767510762478021927, 9.882401166416226415873153865669, 10.61804269149265698780892668657

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