Properties

Degree 4
Conductor $ 7 \cdot 61 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s − 5-s − 4·9-s + 2·10-s + 11-s + 3·13-s + 16-s + 8·18-s + 2·19-s − 20-s − 2·22-s + 23-s + 5·25-s − 6·26-s − 4·29-s − 6·31-s + 2·32-s − 4·36-s − 10·37-s − 4·38-s − 3·41-s + 8·43-s + 44-s + 4·45-s − 2·46-s − 6·49-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s − 0.447·5-s − 4/3·9-s + 0.632·10-s + 0.301·11-s + 0.832·13-s + 1/4·16-s + 1.88·18-s + 0.458·19-s − 0.223·20-s − 0.426·22-s + 0.208·23-s + 25-s − 1.17·26-s − 0.742·29-s − 1.07·31-s + 0.353·32-s − 2/3·36-s − 1.64·37-s − 0.648·38-s − 0.468·41-s + 1.21·43-s + 0.150·44-s + 0.596·45-s − 0.294·46-s − 6/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(427\)    =    \(7 \cdot 61\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{427} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 427,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.1899303686$
$L(\frac12)$  $\approx$  $0.1899303686$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;61\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;61\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 8 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$V_4$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$V_4$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$D_{4}$ \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$V_4$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$D_{4}$ \( 1 + 5 T - 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 7 T + 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + T - 50 T^{2} + p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−19.7584408161, −19.04172481, −18.5531796814, −18.1468454152, −17.3389199371, −17.0838002888, −16.3856158183, −15.7051202259, −14.9330210292, −14.2544394695, −13.6313060333, −12.6120433993, −11.8661381446, −11.0966840919, −10.6271890885, −9.43105940781, −8.96729298284, −8.42134292864, −7.61748176151, −6.43565236949, −5.29476270386, −3.45243698121, 3.45243698121, 5.29476270386, 6.43565236949, 7.61748176151, 8.42134292864, 8.96729298284, 9.43105940781, 10.6271890885, 11.0966840919, 11.8661381446, 12.6120433993, 13.6313060333, 14.2544394695, 14.9330210292, 15.7051202259, 16.3856158183, 17.0838002888, 17.3389199371, 18.1468454152, 18.5531796814, 19.04172481, 19.7584408161

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