Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 3·16-s − 8·19-s − 5·25-s + 16·31-s + 14·49-s + 4·61-s + 7·64-s + 8·76-s − 32·79-s + 5·100-s + 28·109-s − 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1/2·4-s − 3/4·16-s − 1.83·19-s − 25-s + 2.87·31-s + 2·49-s + 0.512·61-s + 7/8·64-s + 0.917·76-s − 3.60·79-s + 1/2·100-s + 2.68·109-s − 2·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2025\)    =    \(3^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2025} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 2025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5869732169$
$L(\frac12)$  $\approx$  $0.5869732169$
$L(\frac{3}{2})$   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 \( 1 \)
5$C_2$ \( 1 + p T^{2} \)
good2$V_4$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$V_4$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$V_4$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$V_4$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$V_4$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$V_4$ \( 1 + 154 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−13.15723379959748243640876161274, −12.96574043565923018826779476430, −11.92843380880703408712681539674, −11.66876266699699984465493727952, −10.71802248997556069743590768513, −10.18166894925267983021659200493, −9.570850536528072839705758819793, −8.529513714199935388900375216623, −8.506199327349381512983870230050, −7.36447877954871795658687378464, −6.52566376180141557577636495203, −5.84873109781954041807523169102, −4.63076666346438307078059690242, −4.10424053282794606423576741067, −2.49439517148849264400889437608, 2.49439517148849264400889437608, 4.10424053282794606423576741067, 4.63076666346438307078059690242, 5.84873109781954041807523169102, 6.52566376180141557577636495203, 7.36447877954871795658687378464, 8.506199327349381512983870230050, 8.529513714199935388900375216623, 9.570850536528072839705758819793, 10.18166894925267983021659200493, 10.71802248997556069743590768513, 11.66876266699699984465493727952, 11.92843380880703408712681539674, 12.96574043565923018826779476430, 13.15723379959748243640876161274

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