Properties

Degree 4
Conductor $ 2^{7} \cdot 7^{3} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 2·9-s − 8·13-s − 14-s + 16-s + 2·18-s − 10·25-s + 8·26-s + 28-s − 8·31-s − 32-s − 2·36-s + 16·43-s − 24·47-s + 49-s + 10·50-s − 8·52-s − 56-s + 16·61-s + 8·62-s − 2·63-s + 64-s − 8·67-s + 2·72-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 2.21·13-s − 0.267·14-s + 1/4·16-s + 0.471·18-s − 2·25-s + 1.56·26-s + 0.188·28-s − 1.43·31-s − 0.176·32-s − 1/3·36-s + 2.43·43-s − 3.50·47-s + 1/7·49-s + 1.41·50-s − 1.10·52-s − 0.133·56-s + 2.04·61-s + 1.01·62-s − 0.251·63-s + 1/8·64-s − 0.977·67-s + 0.235·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(43904\)    =    \(2^{7} \cdot 7^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{43904} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 43904,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
7$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.765547119459919407856461234632, −9.649484617797021810103268663298, −8.934540413043951629895618965043, −8.408228795468761224431793416064, −7.68957389720506083723150978195, −7.57571100088867902110310233811, −6.96175265373852036821224883683, −6.17368924384109906924363362759, −5.57928681742950427486583645839, −5.06701824847582369879735343431, −4.31547623044891473234039816601, −3.42443184461883656887541100921, −2.49739457868022289115900838847, −1.90996831082337002056034743405, 0, 1.90996831082337002056034743405, 2.49739457868022289115900838847, 3.42443184461883656887541100921, 4.31547623044891473234039816601, 5.06701824847582369879735343431, 5.57928681742950427486583645839, 6.17368924384109906924363362759, 6.96175265373852036821224883683, 7.57571100088867902110310233811, 7.68957389720506083723150978195, 8.408228795468761224431793416064, 8.934540413043951629895618965043, 9.649484617797021810103268663298, 9.765547119459919407856461234632

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