Properties

Degree 4
Conductor $ 3 \cdot 461 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s − 5-s + 4·6-s − 6·7-s + 4·8-s + 4·9-s + 2·10-s − 5·13-s + 12·14-s + 2·15-s − 4·16-s + 5·17-s − 8·18-s − 19-s + 12·21-s − 3·23-s − 8·24-s − 2·25-s + 10·26-s − 5·27-s − 2·29-s − 4·30-s − 10·34-s + 6·35-s − 2·37-s + 2·38-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s − 0.447·5-s + 1.63·6-s − 2.26·7-s + 1.41·8-s + 4/3·9-s + 0.632·10-s − 1.38·13-s + 3.20·14-s + 0.516·15-s − 16-s + 1.21·17-s − 1.88·18-s − 0.229·19-s + 2.61·21-s − 0.625·23-s − 1.63·24-s − 2/5·25-s + 1.96·26-s − 0.962·27-s − 0.371·29-s − 0.730·30-s − 1.71·34-s + 1.01·35-s − 0.328·37-s + 0.324·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1383 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1383 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1383\)    =    \(3 \cdot 461\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1383} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1383,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;461\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;461\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
461$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 12 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T - 15 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$D_{4}$ \( 1 + T + 17 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T - 55 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 9 T + 114 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T - 6 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T - 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 9 T + 116 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 13 T + 69 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
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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

−19.4188944367, −19.1191049798, −18.6290127978, −18.1593597947, −17.4959716521, −17.0108454244, −16.5757315954, −16.2390668987, −15.6793024061, −14.9315636556, −13.9897300172, −13.1929983553, −12.7079088414, −12.2260218054, −11.6288219935, −10.3949471442, −10.1148544081, −9.6837731093, −9.23477453625, −8.12662421762, −7.38077067158, −6.7212560813, −5.8263005439, −4.73620779106, −3.53481544437, 0, 3.53481544437, 4.73620779106, 5.8263005439, 6.7212560813, 7.38077067158, 8.12662421762, 9.23477453625, 9.6837731093, 10.1148544081, 10.3949471442, 11.6288219935, 12.2260218054, 12.7079088414, 13.1929983553, 13.9897300172, 14.9315636556, 15.6793024061, 16.2390668987, 16.5757315954, 17.0108454244, 17.4959716521, 18.1593597947, 18.6290127978, 19.1191049798, 19.4188944367

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