Properties

Degree 4
Conductor $ 3 \cdot 349 $
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-s − 4-s + 2·5-s − 2·7-s + 8-s − 2·9-s − 2·10-s + 2·11-s − 2·13-s + 2·14-s − 16-s − 5·17-s + 2·18-s − 19-s − 2·20-s − 2·22-s + 9·23-s − 2·25-s + 2·26-s + 3·27-s + 2·28-s + 3·29-s − 3·31-s + 5·32-s + 5·34-s − 4·35-s + 2·36-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s − 2/3·9-s − 0.632·10-s + 0.603·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s − 1.21·17-s + 0.471·18-s − 0.229·19-s − 0.447·20-s − 0.426·22-s + 1.87·23-s − 2/5·25-s + 0.392·26-s + 0.577·27-s + 0.377·28-s + 0.557·29-s − 0.538·31-s + 0.883·32-s + 0.857·34-s − 0.676·35-s + 1/3·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1047 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1047 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1047\)    =    \(3 \cdot 349\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1047} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1047,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3564335969$
$L(\frac12)$  $\approx$  $0.3564335969$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;349\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;349\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
349$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 10 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$V_4$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \)
29$D_{4}$ \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 42 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$D_{4}$ \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$V_4$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 168 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 11 T + 114 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 218 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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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

−19.7968521301, −19.2249235126, −18.9453242869, −17.9903099784, −17.7293431072, −17.2998590508, −16.8055644698, −16.1750262029, −15.3559560164, −14.7777463264, −14.0597934438, −13.5558909921, −13.0067565854, −12.3738832875, −11.4474875405, −10.8055161856, −9.98439800696, −9.30637594117, −9.04493976381, −8.38384667589, −7.0651860827, −6.45778544372, −5.48737059938, −4.3855980388, −2.73820674806, 2.73820674806, 4.3855980388, 5.48737059938, 6.45778544372, 7.0651860827, 8.38384667589, 9.04493976381, 9.30637594117, 9.98439800696, 10.8055161856, 11.4474875405, 12.3738832875, 13.0067565854, 13.5558909921, 14.0597934438, 14.7777463264, 15.3559560164, 16.1750262029, 16.8055644698, 17.2998590508, 17.7293431072, 17.9903099784, 18.9453242869, 19.2249235126, 19.7968521301

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