Properties

Degree 4
Conductor $ 2^{8} \cdot 3 \cdot 331 $
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-s + 5-s + 2·11-s + 2·13-s − 15-s − 7·17-s + 5·19-s + 25-s + 4·27-s − 10·29-s − 5·31-s − 2·33-s − 10·37-s − 2·39-s + 2·41-s − 3·43-s − 8·47-s − 10·49-s + 7·51-s − 5·53-s + 2·55-s − 5·57-s − 2·59-s − 4·61-s + 2·65-s − 13·67-s + 71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.603·11-s + 0.554·13-s − 0.258·15-s − 1.69·17-s + 1.14·19-s + 1/5·25-s + 0.769·27-s − 1.85·29-s − 0.898·31-s − 0.348·33-s − 1.64·37-s − 0.320·39-s + 0.312·41-s − 0.457·43-s − 1.16·47-s − 1.42·49-s + 0.980·51-s − 0.686·53-s + 0.269·55-s − 0.662·57-s − 0.260·59-s − 0.512·61-s + 0.248·65-s − 1.58·67-s + 0.118·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 254208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 254208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(254208\)    =    \(2^{8} \cdot 3 \cdot 331\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{254208} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 254208,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;331\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;331\}$, 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$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
331$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 20 T + p T^{2} ) \)
good5$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 5 T + 46 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 3 T + 42 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 13 T + 130 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 21 T + 230 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T - 2 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−13.4962167111, −13.0203538992, −12.5856475638, −12.1414143376, −11.734051894, −11.1407354285, −10.977867312, −10.7762072281, −9.93204844465, −9.56785994281, −9.20532264973, −8.80731187151, −8.30909187397, −7.72779837778, −7.10968471204, −6.74220283866, −6.34616686967, −5.79968090222, −5.28410697971, −4.86429399519, −4.21779112833, −3.52562832638, −3.06054333309, −1.94532391276, −1.5036871884, 0, 1.5036871884, 1.94532391276, 3.06054333309, 3.52562832638, 4.21779112833, 4.86429399519, 5.28410697971, 5.79968090222, 6.34616686967, 6.74220283866, 7.10968471204, 7.72779837778, 8.30909187397, 8.80731187151, 9.20532264973, 9.56785994281, 9.93204844465, 10.7762072281, 10.977867312, 11.1407354285, 11.734051894, 12.1414143376, 12.5856475638, 13.0203538992, 13.4962167111

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