Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 31 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 2·5-s − 3·7-s − 12-s + 13-s + 2·15-s + 16-s + 3·19-s − 2·20-s + 3·21-s − 2·23-s + 3·25-s + 27-s − 3·28-s − 6·31-s + 6·35-s + 3·37-s − 39-s + 14·41-s − 10·43-s + 2·47-s − 48-s + 5·49-s + 52-s − 4·53-s − 3·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 0.894·5-s − 1.13·7-s − 0.288·12-s + 0.277·13-s + 0.516·15-s + 1/4·16-s + 0.688·19-s − 0.447·20-s + 0.654·21-s − 0.417·23-s + 3/5·25-s + 0.192·27-s − 0.566·28-s − 1.07·31-s + 1.01·35-s + 0.493·37-s − 0.160·39-s + 2.18·41-s − 1.52·43-s + 0.291·47-s − 0.144·48-s + 5/7·49-s + 0.138·52-s − 0.549·53-s − 0.397·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1116\)    =    \(2^{2} \cdot 3^{2} \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1116} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1116,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.4354897184\)
\(L(\frac12)\)  \(\approx\)  \(0.4354897184\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;31\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + T + T^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 7 T + p T^{2} ) \)
good5$D_{4}$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$D_{4}$ \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 7 T + 62 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
89$D_{4}$ \( 1 + 8 T + 16 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−19.8269873343, −19.2883757539, −18.8516756234, −18.0948767073, −17.7136828040, −16.6192987778, −16.5175014031, −15.9517998857, −15.5242386030, −14.7983059252, −14.1647667290, −13.2277394012, −12.7370138563, −12.1259293035, −11.4968156485, −11.0450555869, −10.2451872085, −9.55019977604, −8.71792977509, −7.72875675432, −7.10805164474, −6.27604807303, −5.52797435783, −4.15236732343, −3.09412679935, 3.09412679935, 4.15236732343, 5.52797435783, 6.27604807303, 7.10805164474, 7.72875675432, 8.71792977509, 9.55019977604, 10.2451872085, 11.0450555869, 11.4968156485, 12.1259293035, 12.7370138563, 13.2277394012, 14.1647667290, 14.7983059252, 15.5242386030, 15.9517998857, 16.5175014031, 16.6192987778, 17.7136828040, 18.0948767073, 18.8516756234, 19.2883757539, 19.8269873343

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