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  − 3-s − 3·5-s + 7-s + 11-s − 8·13-s + 3·15-s − 4·17-s − 21-s − 8·23-s + 5·25-s + 27-s − 14·29-s − 11·31-s − 33-s − 3·35-s − 4·37-s + 8·39-s − 8·41-s − 4·43-s + 2·47-s − 6·49-s + 4·51-s + 11·53-s − 3·55-s − 7·59-s − 10·61-s + 24·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 0.377·7-s + 0.301·11-s − 2.21·13-s + 0.774·15-s − 0.970·17-s − 0.218·21-s − 1.66·23-s + 25-s + 0.192·27-s − 2.59·29-s − 1.97·31-s − 0.174·33-s − 0.507·35-s − 0.657·37-s + 1.28·39-s − 1.24·41-s − 0.609·43-s + 0.291·47-s − 6/7·49-s + 0.560·51-s + 1.51·53-s − 0.404·55-s − 0.911·59-s − 1.28·61-s + 2.97·65-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: induced by $\chi_{672} (1, \cdot )$
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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−10.14281054346396365067246227295, −10.08584248173242507009655876891, −9.401512696234622475758444116524, −8.901249933039359500505315275459, −8.714912023034238691459997068903, −7.78153770049909816015557996524, −7.64519093784188859379067915862, −7.40001818451541070339112726798, −6.84888932591344943018180571278, −6.34664510997124493103410027810, −5.60390831541413157464302935074, −5.28429918352496554247769353546, −4.69026023326786902318418656434, −4.35529245539687864170234320768, −3.63209939211472966796986145547, −3.37361162170638058414250233083, −2.02994942434856337974335766238, −2.01674172808992553288173504584, 0, 0, 2.01674172808992553288173504584, 2.02994942434856337974335766238, 3.37361162170638058414250233083, 3.63209939211472966796986145547, 4.35529245539687864170234320768, 4.69026023326786902318418656434, 5.28429918352496554247769353546, 5.60390831541413157464302935074, 6.34664510997124493103410027810, 6.84888932591344943018180571278, 7.40001818451541070339112726798, 7.64519093784188859379067915862, 7.78153770049909816015557996524, 8.714912023034238691459997068903, 8.901249933039359500505315275459, 9.401512696234622475758444116524, 10.08584248173242507009655876891, 10.14281054346396365067246227295

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