Properties

Degree 4
Conductor $ 2^{6} \cdot 5^{2} \cdot 53^{2} $
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  − 8·7-s − 6·9-s + 8·11-s − 4·13-s + 4·17-s + 25-s − 4·29-s + 12·37-s − 16·43-s + 8·47-s + 34·49-s + 6·53-s − 8·59-s + 48·63-s − 64·77-s + 27·81-s − 12·89-s + 32·91-s − 28·97-s − 48·99-s + 36·113-s + 24·117-s − 32·119-s + 26·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 3.02·7-s − 2·9-s + 2.41·11-s − 1.10·13-s + 0.970·17-s + 1/5·25-s − 0.742·29-s + 1.97·37-s − 2.43·43-s + 1.16·47-s + 34/7·49-s + 0.824·53-s − 1.04·59-s + 6.04·63-s − 7.29·77-s + 3·81-s − 1.27·89-s + 3.35·91-s − 2.84·97-s − 4.82·99-s + 3.38·113-s + 2.21·117-s − 2.93·119-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4494400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4494400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4494400\)    =    \(2^{6} \cdot 5^{2} \cdot 53^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4494400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 4494400,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.6053426183\)
\(L(\frac12)\)  \(\approx\)  \(0.6053426183\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;53\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;53\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
53$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−7.03272073765830561710060363936, −6.97238544391138654785050907556, −6.47620527672720490268023963039, −6.19488475790825972368201105633, −5.74400568682439787270821640474, −5.70093888866808737514223874840, −4.92830317854793887000000437339, −4.20541452241974808193662686957, −3.85660778256709418477363411239, −3.26180587408034592610487247010, −3.17885562764900314427338666343, −2.74391605430030632965486247891, −2.09377714571862097173840160450, −1.05684399493105634109918684829, −0.31235926354071515964561393039, 0.31235926354071515964561393039, 1.05684399493105634109918684829, 2.09377714571862097173840160450, 2.74391605430030632965486247891, 3.17885562764900314427338666343, 3.26180587408034592610487247010, 3.85660778256709418477363411239, 4.20541452241974808193662686957, 4.92830317854793887000000437339, 5.70093888866808737514223874840, 5.74400568682439787270821640474, 6.19488475790825972368201105633, 6.47620527672720490268023963039, 6.97238544391138654785050907556, 7.03272073765830561710060363936

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