Properties

Degree 4
Conductor $ 2 \cdot 3 \cdot 173 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 3·5-s + 2·7-s − 2·8-s − 2·9-s + 2·13-s + 16-s + 4·17-s − 3·19-s + 3·20-s + 23-s + 5·25-s − 3·27-s − 2·28-s − 2·29-s + 5·31-s + 4·32-s − 6·35-s + 2·36-s − 10·37-s + 6·40-s + 9·41-s − 2·43-s + 6·45-s + 47-s + 2·49-s − 2·52-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.34·5-s + 0.755·7-s − 0.707·8-s − 2/3·9-s + 0.554·13-s + 1/4·16-s + 0.970·17-s − 0.688·19-s + 0.670·20-s + 0.208·23-s + 25-s − 0.577·27-s − 0.377·28-s − 0.371·29-s + 0.898·31-s + 0.707·32-s − 1.01·35-s + 1/3·36-s − 1.64·37-s + 0.948·40-s + 1.40·41-s − 0.304·43-s + 0.894·45-s + 0.145·47-s + 2/7·49-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1038 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1038 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1038\)    =    \(2 \cdot 3 \cdot 173\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1038} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1038,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4277040851$
$L(\frac12)$  $\approx$  $0.4277040851$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;173\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;173\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
173$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$D_{4}$ \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
73$D_{4}$ \( 1 + T - 76 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T - 98 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−19.8142266263, −19.3281057573, −18.8662502165, −18.3710324382, −17.7370925855, −17.0893272217, −16.7481023495, −15.6894699482, −15.4863991706, −14.7857458168, −14.2684359608, −13.7259764250, −12.7072548341, −12.2385549547, −11.5980789743, −11.1414034090, −10.3802549358, −9.32803553476, −8.59975714676, −8.14070673343, −7.45960724565, −6.24792263742, −5.30362282238, −4.24249049646, −3.26670926715, 3.26670926715, 4.24249049646, 5.30362282238, 6.24792263742, 7.45960724565, 8.14070673343, 8.59975714676, 9.32803553476, 10.3802549358, 11.1414034090, 11.5980789743, 12.2385549547, 12.7072548341, 13.7259764250, 14.2684359608, 14.7857458168, 15.4863991706, 15.6894699482, 16.7481023495, 17.0893272217, 17.7370925855, 18.3710324382, 18.8662502165, 19.3281057573, 19.8142266263

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