Properties

Degree 4
Conductor $ 17 \cdot 79 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 4·4-s + 6·6-s − 3·7-s − 3·8-s − 9-s − 5·11-s − 8·12-s − 13-s + 9·14-s + 3·16-s + 17-s + 3·18-s + 3·19-s + 6·21-s + 15·22-s − 4·23-s + 6·24-s + 25-s + 3·26-s + 6·27-s − 12·28-s − 29-s + 2·31-s − 6·32-s + 10·33-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s + 2.44·6-s − 1.13·7-s − 1.06·8-s − 1/3·9-s − 1.50·11-s − 2.30·12-s − 0.277·13-s + 2.40·14-s + 3/4·16-s + 0.242·17-s + 0.707·18-s + 0.688·19-s + 1.30·21-s + 3.19·22-s − 0.834·23-s + 1.22·24-s + 1/5·25-s + 0.588·26-s + 1.15·27-s − 2.26·28-s − 0.185·29-s + 0.359·31-s − 1.06·32-s + 1.74·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1343 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1343 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1343\)    =    \(17 \cdot 79\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1343} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1343,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;79\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{17,\;79\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad17$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 7 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 5 T + 13 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 9 T + 53 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
83$D_{4}$ \( 1 - 10 T + 179 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T - 121 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−19.5369643736, −19.0023561692, −18.5114786423, −18.0410982367, −17.6665770122, −17.2781096677, −16.5103606707, −16.2972264621, −15.9004808157, −15.0276046263, −14.2105144413, −13.2666974661, −12.7300310917, −11.9868423685, −11.3608889560, −10.7009840795, −10.0662856123, −9.75470396601, −9.04748273303, −8.10203159812, −7.79250426010, −6.66669234304, −5.90054518905, −5.12522948398, −3.02723489011, 0, 3.02723489011, 5.12522948398, 5.90054518905, 6.66669234304, 7.79250426010, 8.10203159812, 9.04748273303, 9.75470396601, 10.0662856123, 10.7009840795, 11.3608889560, 11.9868423685, 12.7300310917, 13.2666974661, 14.2105144413, 15.0276046263, 15.9004808157, 16.2972264621, 16.5103606707, 17.2781096677, 17.6665770122, 18.0410982367, 18.5114786423, 19.0023561692, 19.5369643736

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