Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{3} \cdot 43^{2} $
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  − 4·3-s + 4-s + 7-s + 6·9-s − 4·12-s + 16-s + 4·19-s − 4·21-s − 10·25-s + 4·27-s + 28-s + 6·36-s + 8·43-s − 4·48-s + 49-s + 12·53-s − 16·57-s + 16·61-s + 6·63-s + 64-s − 8·67-s + 4·73-s + 40·75-s + 4·76-s + 16·79-s − 37·81-s − 4·84-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 0.377·7-s + 2·9-s − 1.15·12-s + 1/4·16-s + 0.917·19-s − 0.872·21-s − 2·25-s + 0.769·27-s + 0.188·28-s + 36-s + 1.21·43-s − 0.577·48-s + 1/7·49-s + 1.64·53-s − 2.11·57-s + 2.04·61-s + 0.755·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s + 4.61·75-s + 0.458·76-s + 1.80·79-s − 4.11·81-s − 0.436·84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2536828 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2536828 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2536828\)    =    \(2^{2} \cdot 7^{3} \cdot 43^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2536828} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 2536828,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9050980217$
$L(\frac12)$  $\approx$  $0.9050980217$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;43\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 - T \)
43$C_2$ \( 1 - 8 T + p T^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−7.57571100088867902110310233811, −7.12281746300407781825619756047, −6.60599462539983023605940406469, −6.29453411307540537433198026576, −5.86890715003417802953513572698, −5.57928681742950427486583645839, −5.23281482767021772685797667357, −4.95908623650608970212309354280, −4.15249236209760350913939025796, −3.95306231241954591666123257481, −3.14642645355656320067847829114, −2.49628429257252141928083234036, −1.90704256758975423597314948267, −1.04561791290653755383549208528, −0.51659027018551281736912882930, 0.51659027018551281736912882930, 1.04561791290653755383549208528, 1.90704256758975423597314948267, 2.49628429257252141928083234036, 3.14642645355656320067847829114, 3.95306231241954591666123257481, 4.15249236209760350913939025796, 4.95908623650608970212309354280, 5.23281482767021772685797667357, 5.57928681742950427486583645839, 5.86890715003417802953513572698, 6.29453411307540537433198026576, 6.60599462539983023605940406469, 7.12281746300407781825619756047, 7.57571100088867902110310233811

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