Properties

Degree 4
Conductor $ 2^{7} \cdot 7^{3} $
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  − 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  =  \(0\)
Selberg data  =  \((4,\ 43904,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.9391169980\)
\(L(\frac12)\)  \(\approx\)  \(0.9391169980\)
\(L(\frac{3}{2})\)   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

−10.39623059414136659793705857031, −9.479912708108790860766004869191, −9.205467104844532229657218112988, −8.624667079434487291694904422005, −8.328602349022309226048251750698, −7.55563465371540871894331307120, −7.29200576171944643329126204500, −6.12986038892916091786195943044, −6.04992362572003306544038558924, −5.73625972355331222443538857924, −4.39126934619586232082133319537, −3.87985229263682105648190826671, −3.09628357761679869111062402047, −2.25140513775369021989127014661, −1.00403210582709168291350506092, 1.00403210582709168291350506092, 2.25140513775369021989127014661, 3.09628357761679869111062402047, 3.87985229263682105648190826671, 4.39126934619586232082133319537, 5.73625972355331222443538857924, 6.04992362572003306544038558924, 6.12986038892916091786195943044, 7.29200576171944643329126204500, 7.55563465371540871894331307120, 8.328602349022309226048251750698, 8.624667079434487291694904422005, 9.205467104844532229657218112988, 9.479912708108790860766004869191, 10.39623059414136659793705857031

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