Properties

Degree 4
Conductor $ 2 \cdot 7 \cdot 359 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 4-s − 5·5-s + 8·6-s − 4·7-s + 2·8-s + 8·9-s + 10·10-s − 2·11-s − 4·12-s − 2·13-s + 8·14-s + 20·15-s − 3·16-s − 5·17-s − 16·18-s − 19-s − 5·20-s + 16·21-s + 4·22-s − 2·23-s − 8·24-s + 11·25-s + 4·26-s − 12·27-s − 4·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 2.23·5-s + 3.26·6-s − 1.51·7-s + 0.707·8-s + 8/3·9-s + 3.16·10-s − 0.603·11-s − 1.15·12-s − 0.554·13-s + 2.13·14-s + 5.16·15-s − 3/4·16-s − 1.21·17-s − 3.77·18-s − 0.229·19-s − 1.11·20-s + 3.49·21-s + 0.852·22-s − 0.417·23-s − 1.63·24-s + 11/5·25-s + 0.784·26-s − 2.30·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5026 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5026 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5026\)    =    \(2 \cdot 7 \cdot 359\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5026} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 5026,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;359\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;359\}$, 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$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 3 T + p T^{2} ) \)
359$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 12 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 5 T + 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T - 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
73$D_{4}$ \( 1 - 5 T - 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T - 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 3 T - 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 14 T + 128 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 22 T + 284 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
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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

−17.8495923016, −17.5581062571, −17.0216357563, −16.4670230563, −16.2670847252, −15.7421208447, −15.539509546, −14.8399438075, −13.5428814154, −13.0469494543, −12.393435457, −11.9556808059, −11.632314846, −11.0560794963, −10.4277263074, −10.2134417691, −9.38055330815, −8.62032312509, −7.93450047094, −7.38327125084, −6.68125710006, −6.22569491659, −5.04280368563, −4.48669997652, −3.47675658449, 0, 0, 3.47675658449, 4.48669997652, 5.04280368563, 6.22569491659, 6.68125710006, 7.38327125084, 7.93450047094, 8.62032312509, 9.38055330815, 10.2134417691, 10.4277263074, 11.0560794963, 11.632314846, 11.9556808059, 12.393435457, 13.0469494543, 13.5428814154, 14.8399438075, 15.539509546, 15.7421208447, 16.2670847252, 16.4670230563, 17.0216357563, 17.5581062571, 17.8495923016

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