Properties

Degree 4
Conductor $ 3^{2} \cdot 11149 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 5-s − 3·7-s − 4·8-s + 9-s + 2·10-s − 11-s − 4·13-s − 6·14-s − 4·16-s + 2·17-s + 2·18-s − 2·19-s − 2·22-s + 4·23-s − 8·26-s − 8·29-s − 3·31-s + 4·34-s − 3·35-s − 3·37-s − 4·38-s − 4·40-s − 12·43-s + 45-s + 8·46-s − 2·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.447·5-s − 1.13·7-s − 1.41·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 1.10·13-s − 1.60·14-s − 16-s + 0.485·17-s + 0.471·18-s − 0.458·19-s − 0.426·22-s + 0.834·23-s − 1.56·26-s − 1.48·29-s − 0.538·31-s + 0.685·34-s − 0.507·35-s − 0.493·37-s − 0.648·38-s − 0.632·40-s − 1.82·43-s + 0.149·45-s + 1.17·46-s − 0.291·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100341 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100341 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100341\)    =    \(3^{2} \cdot 11149\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100341} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 100341,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11149\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;11149\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11149$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 17 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
5$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$D_{4}$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 2 T + 28 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - T + 96 T^{2} - p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 9 T + 103 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T - 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 20 T + 238 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 15 T + 179 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 9 T + 122 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$D_{4}$ \( 1 + 3 T - 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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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

−14.3064353073, −13.65962313, −13.2862677791, −13.011600191, −12.792228146, −12.4260051142, −11.7741323785, −11.4930085411, −10.6226012437, −10.2211559978, −9.76456440647, −9.47331797115, −8.87111644047, −8.53686281174, −7.56145262737, −7.22734130136, −6.6494056028, −6.0451953987, −5.49051825632, −5.17020785041, −4.59665457139, −3.95624699813, −3.36557443029, −2.89994015688, −1.88909270043, 0, 1.88909270043, 2.89994015688, 3.36557443029, 3.95624699813, 4.59665457139, 5.17020785041, 5.49051825632, 6.0451953987, 6.6494056028, 7.22734130136, 7.56145262737, 8.53686281174, 8.87111644047, 9.47331797115, 9.76456440647, 10.2211559978, 10.6226012437, 11.4930085411, 11.7741323785, 12.4260051142, 12.792228146, 13.011600191, 13.2862677791, 13.65962313, 14.3064353073

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