Properties

Degree 4
Conductor $ 2^{2} \cdot 2113 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·3-s − 6·5-s + 4·6-s − 3·7-s + 8-s + 7·9-s + 6·10-s − 4·11-s + 2·13-s + 3·14-s + 24·15-s − 16-s − 17-s − 7·18-s − 6·19-s + 12·21-s + 4·22-s − 23-s − 4·24-s + 18·25-s − 2·26-s − 4·27-s − 6·29-s − 24·30-s − 2·31-s + 16·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s − 2.68·5-s + 1.63·6-s − 1.13·7-s + 0.353·8-s + 7/3·9-s + 1.89·10-s − 1.20·11-s + 0.554·13-s + 0.801·14-s + 6.19·15-s − 1/4·16-s − 0.242·17-s − 1.64·18-s − 1.37·19-s + 2.61·21-s + 0.852·22-s − 0.208·23-s − 0.816·24-s + 18/5·25-s − 0.392·26-s − 0.769·27-s − 1.11·29-s − 4.38·30-s − 0.359·31-s + 2.78·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8452 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8452 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(8452\)    =    \(2^{2} \cdot 2113\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8452} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 8452,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;2113\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;2113\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + T^{2} \)
2113$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 34 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$D_{4}$ \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 7 T + 146 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.2236475593, −16.5797390066, −16.3888370194, −16.1117264558, −15.6038907382, −15.233120532, −14.6982031675, −13.3882860932, −12.974992486, −12.5004054022, −11.9829012467, −11.6492727686, −11.0046335153, −10.8317857836, −10.4400187517, −9.51018460387, −8.6884289706, −8.02464277038, −7.66811348541, −6.85547926913, −6.37442807639, −5.6131467912, −4.77809428916, −4.14393091775, −3.31956371415, 0, 0, 3.31956371415, 4.14393091775, 4.77809428916, 5.6131467912, 6.37442807639, 6.85547926913, 7.66811348541, 8.02464277038, 8.6884289706, 9.51018460387, 10.4400187517, 10.8317857836, 11.0046335153, 11.6492727686, 11.9829012467, 12.5004054022, 12.974992486, 13.3882860932, 14.6982031675, 15.233120532, 15.6038907382, 16.1117264558, 16.3888370194, 16.5797390066, 17.2236475593

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