Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{2} \cdot 7^{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 + 5-s + 5·11-s + 15-s − 4·17-s − 8·19-s − 4·23-s + 5·25-s − 27-s − 10·29-s − 3·31-s + 5·33-s + 4·37-s − 4·43-s + 6·47-s − 4·51-s + 9·53-s + 5·55-s − 8·57-s + 11·59-s − 6·61-s − 2·67-s − 4·69-s − 4·71-s + 10·73-s + 5·75-s + 3·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.50·11-s + 0.258·15-s − 0.970·17-s − 1.83·19-s − 0.834·23-s + 25-s − 0.192·27-s − 1.85·29-s − 0.538·31-s + 0.870·33-s + 0.657·37-s − 0.609·43-s + 0.875·47-s − 0.560·51-s + 1.23·53-s + 0.674·55-s − 1.05·57-s + 1.43·59-s − 0.768·61-s − 0.244·67-s − 0.481·69-s − 0.474·71-s + 1.17·73-s + 0.577·75-s + 0.337·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2352} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.506189173\)
\(L(\frac12)\)  \(\approx\)  \(2.506189173\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.264099594723781224880489550601, −8.674904681914837763667206653583, −8.623122995027749086326652796862, −8.210204336131024636060434124044, −7.62889102428376101164693166654, −7.07484583946968114303913920671, −6.98476033230961536257573073360, −6.49128971882048352704348200360, −5.97351072899392792649825708485, −5.90446912952814723134505483135, −5.25456993932562790526417908683, −4.67331392305461226897948667618, −4.24147627282244726410142343926, −3.80262749528093413368229290515, −3.73696130146704588161860501122, −2.81322368978783275942551703448, −2.38084421049568660477990229632, −1.89649472878622797180207758959, −1.53166094255051119392166571361, −0.49950331082421080325331240143, 0.49950331082421080325331240143, 1.53166094255051119392166571361, 1.89649472878622797180207758959, 2.38084421049568660477990229632, 2.81322368978783275942551703448, 3.73696130146704588161860501122, 3.80262749528093413368229290515, 4.24147627282244726410142343926, 4.67331392305461226897948667618, 5.25456993932562790526417908683, 5.90446912952814723134505483135, 5.97351072899392792649825708485, 6.49128971882048352704348200360, 6.98476033230961536257573073360, 7.07484583946968114303913920671, 7.62889102428376101164693166654, 8.210204336131024636060434124044, 8.623122995027749086326652796862, 8.674904681914837763667206653583, 9.264099594723781224880489550601

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