Properties

Degree $4$
Conductor $22500$
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

Degree: \(4\)
Conductor: \(22500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{22500} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 22500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9866916655\)
\(L(\frac12)\) \(\approx\) \(0.9866916655\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.06356630052318700046867394716, −10.08927395051771743893684804550, −9.930723730317650982781533521356, −9.150350739841772685069628447120, −8.696166163424621427674216809504, −8.071729218562685355881418106440, −7.34146793172562160046139993443, −6.60865865069333408239607336365, −6.35306147256574072432179047412, −5.66248146017224201443960953748, −5.34945783064988058695168525193, −4.04063298891995369242558301329, −3.29699975126484092788987476482, −2.90142231580611624816334042439, −1.13669044249807591949724917468, 1.13669044249807591949724917468, 2.90142231580611624816334042439, 3.29699975126484092788987476482, 4.04063298891995369242558301329, 5.34945783064988058695168525193, 5.66248146017224201443960953748, 6.35306147256574072432179047412, 6.60865865069333408239607336365, 7.34146793172562160046139993443, 8.071729218562685355881418106440, 8.696166163424621427674216809504, 9.150350739841772685069628447120, 9.930723730317650982781533521356, 10.08927395051771743893684804550, 11.06356630052318700046867394716

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