Properties

Degree 4
Conductor $ 2^{7} \cdot 3 \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 4·9-s − 7·11-s − 2·12-s + 16-s − 6·17-s − 4·18-s − 4·19-s + 7·22-s + 2·24-s + 4·25-s − 5·27-s − 32-s + 14·33-s + 6·34-s + 4·36-s + 4·38-s − 11·41-s − 43-s − 7·44-s − 2·48-s − 11·49-s − 4·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 4/3·9-s − 2.11·11-s − 0.577·12-s + 1/4·16-s − 1.45·17-s − 0.942·18-s − 0.917·19-s + 1.49·22-s + 0.408·24-s + 4/5·25-s − 0.962·27-s − 0.176·32-s + 2.43·33-s + 1.02·34-s + 2/3·36-s + 0.648·38-s − 1.71·41-s − 0.152·43-s − 1.05·44-s − 0.288·48-s − 1.57·49-s − 0.565·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(15744\)    =    \(2^{7} \cdot 3 \cdot 41\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{15744} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 15744,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;41\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;41\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 12 T + p T^{2} ) \)
good5$V_4$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
7$V_4$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$V_4$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$V_4$ \( 1 + 33 T^{2} + p^{2} T^{4} \)
29$V_4$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$V_4$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
37$V_4$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
47$V_4$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
53$V_4$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$V_4$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$V_4$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
79$V_4$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.84029278903557493797460719162, −10.46750903863308980307392268619, −9.894091456753804767688149230613, −9.304367627546415876073709126418, −8.399503040276540200170274749395, −8.187649934932811790541702864723, −7.35993356036733808755043340184, −6.71652536738266069584465467093, −6.42162768793371288659243844033, −5.42732003445070923307315686624, −4.99489014438729461108704082071, −4.29854184095109999611534361966, −2.95528667143029691869062466521, −1.94877540993939406225847659283, 0, 1.94877540993939406225847659283, 2.95528667143029691869062466521, 4.29854184095109999611534361966, 4.99489014438729461108704082071, 5.42732003445070923307315686624, 6.42162768793371288659243844033, 6.71652536738266069584465467093, 7.35993356036733808755043340184, 8.187649934932811790541702864723, 8.399503040276540200170274749395, 9.304367627546415876073709126418, 9.894091456753804767688149230613, 10.46750903863308980307392268619, 10.84029278903557493797460719162

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