Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{6} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 13-s + 16-s + 4·19-s + 8·25-s − 8·31-s − 5·37-s + 7·43-s − 6·49-s + 52-s + 7·61-s + 64-s − 17·67-s + 13·73-s + 4·76-s − 11·79-s − 20·97-s + 8·100-s − 8·103-s − 14·109-s − 4·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.277·13-s + 1/4·16-s + 0.917·19-s + 8/5·25-s − 1.43·31-s − 0.821·37-s + 1.06·43-s − 6/7·49-s + 0.138·52-s + 0.896·61-s + 1/8·64-s − 2.07·67-s + 1.52·73-s + 0.458·76-s − 1.23·79-s − 2.03·97-s + 4/5·100-s − 0.788·103-s − 1.34·109-s − 0.363·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.410·148-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20412 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20412 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(20412\)    =    \(2^{2} \cdot 3^{6} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{20412} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 20412,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.321137289$
$L(\frac12)$  $\approx$  $1.321137289$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
good5$V_4$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$V_4$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$V_4$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$V_4$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$V_4$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
47$V_4$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
53$V_4$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$V_4$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$V_4$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−10.78054444854833913410674012940, −10.55376540537036846977385905472, −9.731817561821762417591946518018, −9.258065637262344174182183234024, −8.727186239206226935472447794602, −8.103965047141443897493173295234, −7.43511443565168193777003995180, −6.98507294544779013933608971902, −6.43427702798258515878327163861, −5.58223271934365262380028520310, −5.19338692173146828831048630873, −4.24264059355529231234305244599, −3.40312992704562422441566083600, −2.68585712044940757410764993903, −1.44784256710938688824185178465, 1.44784256710938688824185178465, 2.68585712044940757410764993903, 3.40312992704562422441566083600, 4.24264059355529231234305244599, 5.19338692173146828831048630873, 5.58223271934365262380028520310, 6.43427702798258515878327163861, 6.98507294544779013933608971902, 7.43511443565168193777003995180, 8.103965047141443897493173295234, 8.727186239206226935472447794602, 9.258065637262344174182183234024, 9.731817561821762417591946518018, 10.55376540537036846977385905472, 10.78054444854833913410674012940

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