Properties

Degree 4
Conductor 997
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 − 3·3-s − 2·5-s + 6·6-s − 2·7-s + 4·8-s + 4·9-s + 4·10-s + 4·11-s − 8·13-s + 4·14-s + 6·15-s − 4·16-s − 8·18-s + 19-s + 6·21-s − 8·22-s − 2·23-s − 12·24-s + 16·26-s − 6·27-s − 29-s − 12·30-s − 3·31-s − 12·33-s + 4·35-s − 3·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s − 0.894·5-s + 2.44·6-s − 0.755·7-s + 1.41·8-s + 4/3·9-s + 1.26·10-s + 1.20·11-s − 2.21·13-s + 1.06·14-s + 1.54·15-s − 16-s − 1.88·18-s + 0.229·19-s + 1.30·21-s − 1.70·22-s − 0.417·23-s − 2.44·24-s + 3.13·26-s − 1.15·27-s − 0.185·29-s − 2.19·30-s − 0.538·31-s − 2.08·33-s + 0.676·35-s − 0.493·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(997\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{997} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 997,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 997$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 997$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad997$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 28 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$V_4$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 11 T + 88 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$D_{4}$ \( 1 - 5 T + 85 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$D_{4}$ \( 1 - 11 T + 110 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 13 T + 82 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 9 T + 178 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−19.8333740009, −19.4536134475, −19.0956890151, −18.62903691, −17.7704377714, −17.4898655597, −17.0043095466, −16.7781728545, −16.1762197636, −15.4494525594, −14.6717977011, −13.9833649039, −12.9703237378, −12.2511059659, −11.9025998913, −11.3527015429, −10.4576570026, −9.82863844824, −9.43629139792, −8.6234148321, −7.50247649292, −7.12754538244, −6.00294768365, −4.99733294411, −4.05290407737, 0, 4.05290407737, 4.99733294411, 6.00294768365, 7.12754538244, 7.50247649292, 8.6234148321, 9.43629139792, 9.82863844824, 10.4576570026, 11.3527015429, 11.9025998913, 12.2511059659, 12.9703237378, 13.9833649039, 14.6717977011, 15.4494525594, 16.1762197636, 16.7781728545, 17.0043095466, 17.4898655597, 17.7704377714, 18.62903691, 19.0956890151, 19.4536134475, 19.8333740009

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