Properties

Degree $4$
Conductor $451584$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 9-s − 8·11-s − 4·13-s + 2·25-s − 16·31-s − 8·43-s − 4·45-s − 7·49-s + 32·55-s − 4·61-s + 16·65-s + 8·67-s + 81-s − 8·99-s − 36·101-s − 32·103-s + 24·107-s + 36·113-s − 4·117-s + 26·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + ⋯
L(s)  = 1  − 1.78·5-s + 1/3·9-s − 2.41·11-s − 1.10·13-s + 2/5·25-s − 2.87·31-s − 1.21·43-s − 0.596·45-s − 49-s + 4.31·55-s − 0.512·61-s + 1.98·65-s + 0.977·67-s + 1/9·81-s − 0.804·99-s − 3.58·101-s − 3.15·103-s + 2.32·107-s + 3.38·113-s − 0.369·117-s + 2.36·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{451584} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.173282349834764667604075089133, −7.57346950816982656575487446761, −7.26958585488936859387169650645, −7.22685430203741355342227556898, −6.28266974817041814701199375242, −5.59620622280165428169903914653, −5.01357758335939619070346679818, −5.00670845718665013316471514453, −4.12187187931469968184698878478, −3.71533601983866002659839530844, −3.14826068230388029805668446658, −2.50123083621432021396021244496, −1.80970977960128750436372604769, 0, 0, 1.80970977960128750436372604769, 2.50123083621432021396021244496, 3.14826068230388029805668446658, 3.71533601983866002659839530844, 4.12187187931469968184698878478, 5.00670845718665013316471514453, 5.01357758335939619070346679818, 5.59620622280165428169903914653, 6.28266974817041814701199375242, 7.22685430203741355342227556898, 7.26958585488936859387169650645, 7.57346950816982656575487446761, 8.173282349834764667604075089133

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