Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{4} $
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-s + 4-s + 4·7-s − 2·9-s + 12-s − 8·13-s + 16-s + 10·19-s + 4·21-s − 5·27-s + 4·28-s + 4·31-s − 2·36-s + 4·37-s − 8·39-s − 8·43-s + 48-s − 2·49-s − 8·52-s + 10·57-s + 4·61-s − 8·63-s + 64-s − 26·67-s + 22·73-s + 10·76-s − 20·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 1.51·7-s − 2/3·9-s + 0.288·12-s − 2.21·13-s + 1/4·16-s + 2.29·19-s + 0.872·21-s − 0.962·27-s + 0.755·28-s + 0.718·31-s − 1/3·36-s + 0.657·37-s − 1.28·39-s − 1.21·43-s + 0.144·48-s − 2/7·49-s − 1.10·52-s + 1.32·57-s + 0.512·61-s − 1.00·63-s + 1/8·64-s − 3.17·67-s + 2.57·73-s + 1.14·76-s − 2.25·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(22500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{22500} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 22500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.644486109$
$L(\frac12)$  $\approx$  $1.644486109$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 - T + p T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.81431448769284119152476376034, −10.18789957492033670124802464882, −9.481094579683641415186193216160, −9.447991916742081591691197868932, −8.316812877075225116243141130520, −8.114692805267148178564014319106, −7.40487753845237746133032109213, −7.29853134793764088505009214313, −6.28477478517999960844504626470, −5.23380204015925208976616676091, −5.20592148788576143134616479765, −4.36763710614752510700235050554, −3.13165277251593234405465211013, −2.65155040417235216240951299295, −1.63862417385241407780156087735, 1.63862417385241407780156087735, 2.65155040417235216240951299295, 3.13165277251593234405465211013, 4.36763710614752510700235050554, 5.20592148788576143134616479765, 5.23380204015925208976616676091, 6.28477478517999960844504626470, 7.29853134793764088505009214313, 7.40487753845237746133032109213, 8.114692805267148178564014319106, 8.316812877075225116243141130520, 9.447991916742081591691197868932, 9.481094579683641415186193216160, 10.18789957492033670124802464882, 10.81431448769284119152476376034

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