Properties

Degree 4
Conductor $ 3^{3} \cdot 47 $
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 + 3-s − 4-s + 5-s − 6-s − 3·7-s + 8-s + 9-s − 10-s − 3·11-s − 12-s + 3·14-s + 15-s − 16-s + 4·17-s − 18-s + 2·19-s − 20-s − 3·21-s + 3·22-s − 5·23-s + 24-s + 25-s + 27-s + 3·28-s − 3·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.288·12-s + 0.801·14-s + 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.654·21-s + 0.639·22-s − 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.566·28-s − 0.557·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1269 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1269 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1269\)    =    \(3^{3} \cdot 47\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1269} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1269,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4110304678$
$L(\frac12)$  $\approx$  $0.4110304678$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;47\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;47\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( 1 - T \)
47$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 8 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$D_{4}$ \( 1 - 7 T + 16 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
61$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
83$V_4$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 5 T + 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
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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

−19.4568642495, −18.9557822006, −18.3915669991, −18.2083615186, −17.5883218457, −16.7687059991, −16.3481021084, −15.8247173824, −15.1395894603, −14.4729157893, −13.7211064669, −13.4260934202, −12.763572476, −12.2136120398, −11.2365089938, −10.1880140279, −9.91785898148, −9.40018764063, −8.71149648676, −7.96340936718, −7.26931086003, −6.17378930477, −5.34879374046, −3.97282357485, −2.75360774865, 2.75360774865, 3.97282357485, 5.34879374046, 6.17378930477, 7.26931086003, 7.96340936718, 8.71149648676, 9.40018764063, 9.91785898148, 10.1880140279, 11.2365089938, 12.2136120398, 12.763572476, 13.4260934202, 13.7211064669, 14.4729157893, 15.1395894603, 15.8247173824, 16.3481021084, 16.7687059991, 17.5883218457, 18.2083615186, 18.3915669991, 18.9557822006, 19.4568642495

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