Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{6} \cdot 7^{2} $
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 + 7-s + 9·13-s + 16-s + 8·25-s − 28-s + 5·37-s − 7·43-s − 6·49-s − 9·52-s + 9·61-s − 64-s − 17·67-s − 9·73-s − 11·79-s + 9·91-s − 8·100-s + 14·109-s + 112-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.377·7-s + 2.49·13-s + 1/4·16-s + 8/5·25-s − 0.188·28-s + 0.821·37-s − 1.06·43-s − 6/7·49-s − 1.24·52-s + 1.15·61-s − 1/8·64-s − 2.07·67-s − 1.05·73-s − 1.23·79-s + 0.943·91-s − 4/5·100-s + 1.34·109-s + 0.0944·112-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.410·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(142884\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{142884} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 142884,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.729774751$
$L(\frac12)$  $\approx$  $1.729774751$
$L(\frac{3}{2})$   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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$V_4$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$V_4$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$V_4$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$ \( ( 1 - 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.139653344749279964318027512398, −8.752773473131153470061873036103, −8.460607111207407981112349907621, −8.076683354683520637597902581608, −7.40572156925869950955407390609, −6.78347510071548608962964563042, −6.31331610173668692333696289993, −5.82067067992331099005958496359, −5.31163955771378476686658667715, −4.53748311300456991048958963030, −4.22307191307728643314985931790, −3.37658620712098305101864082824, −3.01922594705373137377702138338, −1.72715717203838961223003097116, −1.01441521593897187512941634728, 1.01441521593897187512941634728, 1.72715717203838961223003097116, 3.01922594705373137377702138338, 3.37658620712098305101864082824, 4.22307191307728643314985931790, 4.53748311300456991048958963030, 5.31163955771378476686658667715, 5.82067067992331099005958496359, 6.31331610173668692333696289993, 6.78347510071548608962964563042, 7.40572156925869950955407390609, 8.076683354683520637597902581608, 8.460607111207407981112349907621, 8.752773473131153470061873036103, 9.139653344749279964318027512398

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