Properties

Degree 4
Conductor $ 3 \cdot 359 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 3·7-s − 11-s + 13-s − 4·15-s − 4·16-s + 2·17-s − 19-s + 6·21-s + 4·23-s − 2·25-s + 5·27-s + 6·29-s − 4·31-s + 2·33-s − 6·35-s + 9·37-s − 2·39-s + 6·41-s + 2·43-s + 2·47-s + 8·48-s + 3·49-s − 4·51-s − 5·53-s − 2·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 1.13·7-s − 0.301·11-s + 0.277·13-s − 1.03·15-s − 16-s + 0.485·17-s − 0.229·19-s + 1.30·21-s + 0.834·23-s − 2/5·25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s − 1.01·35-s + 1.47·37-s − 0.320·39-s + 0.937·41-s + 0.304·43-s + 0.291·47-s + 1.15·48-s + 3/7·49-s − 0.560·51-s − 0.686·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1077\)    =    \(3 \cdot 359\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1077} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1077,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4062914313$
$L(\frac12)$  $\approx$  $0.4062914313$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;359\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;359\}$, 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$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
359$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 30 T + p T^{2} ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$D_{4}$ \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
31$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T - 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$D_{4}$ \( 1 + 5 T + 75 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 160 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_4$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 3 T - 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 78 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 5 T + 3 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−19.7885707311, −19.3350967233, −18.6069266154, −18.0063475734, −17.7391719162, −17.0104791184, −16.6066332273, −16.1033269463, −15.580196108, −14.7151030466, −13.9721229615, −13.4012859044, −12.8795356592, −12.2404190603, −11.5432279366, −10.8230715911, −10.392272623, −9.40796104497, −9.14344422481, −7.8726125107, −6.77845005959, −6.13876455308, −5.6716154469, −4.56188709982, −2.87034808619, 2.87034808619, 4.56188709982, 5.6716154469, 6.13876455308, 6.77845005959, 7.8726125107, 9.14344422481, 9.40796104497, 10.392272623, 10.8230715911, 11.5432279366, 12.2404190603, 12.8795356592, 13.4012859044, 13.9721229615, 14.7151030466, 15.580196108, 16.1033269463, 16.6066332273, 17.0104791184, 17.7391719162, 18.0063475734, 18.6069266154, 19.3350967233, 19.7885707311

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