Properties

Degree 4
Conductor $ 2^{4} \cdot 3 \cdot 23 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 2·9-s + 2·11-s + 13-s − 2·17-s + 2·19-s + 3·23-s + 6·25-s + 3·27-s − 29-s + 3·31-s − 12·37-s + 41-s − 4·43-s + 8·45-s + 3·47-s + 10·49-s − 4·53-s − 8·55-s − 8·59-s − 8·61-s − 4·65-s + 2·67-s + 7·71-s + 11·73-s − 10·79-s + 4·81-s + ⋯
L(s)  = 1  − 1.78·5-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.485·17-s + 0.458·19-s + 0.625·23-s + 6/5·25-s + 0.577·27-s − 0.185·29-s + 0.538·31-s − 1.97·37-s + 0.156·41-s − 0.609·43-s + 1.19·45-s + 0.437·47-s + 10/7·49-s − 0.549·53-s − 1.07·55-s − 1.04·59-s − 1.02·61-s − 0.496·65-s + 0.244·67-s + 0.830·71-s + 1.28·73-s − 1.12·79-s + 4/9·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1104} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1104,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4453748542$
$L(\frac12)$  $\approx$  $0.4453748542$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;23\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$V_4$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
53$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$D_{4}$ \( 1 - 11 T + 124 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−19.8278713562, −19.4149334317, −18.8035573521, −18.3822107134, −17.442504323, −17.0875875799, −16.3828873887, −15.7665595893, −15.3664104464, −14.9256317224, −13.9745060942, −13.7262317709, −12.5219615781, −12.1574493503, −11.5467710813, −11.077253044, −10.3856958944, −9.19706065176, −8.66273573031, −7.97621904415, −7.2294207085, −6.45160845933, −5.18110661744, −4.13114025718, −3.25020887365, 3.25020887365, 4.13114025718, 5.18110661744, 6.45160845933, 7.2294207085, 7.97621904415, 8.66273573031, 9.19706065176, 10.3856958944, 11.077253044, 11.5467710813, 12.1574493503, 12.5219615781, 13.7262317709, 13.9745060942, 14.9256317224, 15.3664104464, 15.7665595893, 16.3828873887, 17.0875875799, 17.442504323, 18.3822107134, 18.8035573521, 19.4149334317, 19.8278713562

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