Properties

Degree 4
Conductor $ 3^{3} \cdot 5^{2} \cdot 7^{3} $
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-s − 3·4-s − 2·5-s + 7-s + 9-s + 3·12-s + 2·15-s + 5·16-s − 4·17-s + 6·20-s − 21-s + 3·25-s − 27-s − 3·28-s − 2·35-s − 3·36-s − 4·37-s + 12·41-s + 8·43-s − 2·45-s − 16·47-s − 5·48-s + 49-s + 4·51-s − 8·59-s − 6·60-s + 63-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.866·12-s + 0.516·15-s + 5/4·16-s − 0.970·17-s + 1.34·20-s − 0.218·21-s + 3/5·25-s − 0.192·27-s − 0.566·28-s − 0.338·35-s − 1/2·36-s − 0.657·37-s + 1.87·41-s + 1.21·43-s − 0.298·45-s − 2.33·47-s − 0.721·48-s + 1/7·49-s + 0.560·51-s − 1.04·59-s − 0.774·60-s + 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(231525\)    =    \(3^{3} \cdot 5^{2} \cdot 7^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{231525} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 231525,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( 1 + T \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.843078054411908501657519136744, −8.261177977090796325600765542566, −7.73831918709431562684545235915, −7.69164818258356184893378968692, −6.67009631001529545506185276841, −6.48451110526696282246961711325, −5.69208011630684437322158261218, −5.01564976002989168468776414543, −4.87484917945516723546545158294, −4.17084275561163201389021211529, −3.94068846921577588199342611811, −3.18167275817234212778601509206, −2.17599002568151719086793905114, −0.982469269817440515290174005672, 0, 0.982469269817440515290174005672, 2.17599002568151719086793905114, 3.18167275817234212778601509206, 3.94068846921577588199342611811, 4.17084275561163201389021211529, 4.87484917945516723546545158294, 5.01564976002989168468776414543, 5.69208011630684437322158261218, 6.48451110526696282246961711325, 6.67009631001529545506185276841, 7.69164818258356184893378968692, 7.73831918709431562684545235915, 8.261177977090796325600765542566, 8.843078054411908501657519136744

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