Properties

Degree 4
Conductor $ 3^{3} \cdot 61 $
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-s − 3·3-s − 2·4-s + 3·6-s − 6·7-s + 3·8-s + 6·9-s + 2·11-s + 6·12-s − 3·13-s + 6·14-s + 16-s + 17-s − 6·18-s + 18·21-s − 2·22-s − 8·23-s − 9·24-s − 25-s + 3·26-s − 9·27-s + 12·28-s + 2·29-s − 2·32-s − 6·33-s − 34-s − 12·36-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 4-s + 1.22·6-s − 2.26·7-s + 1.06·8-s + 2·9-s + 0.603·11-s + 1.73·12-s − 0.832·13-s + 1.60·14-s + 1/4·16-s + 0.242·17-s − 1.41·18-s + 3.92·21-s − 0.426·22-s − 1.66·23-s − 1.83·24-s − 1/5·25-s + 0.588·26-s − 1.73·27-s + 2.26·28-s + 0.371·29-s − 0.353·32-s − 1.04·33-s − 0.171·34-s − 2·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1647 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1647 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1647\)    =    \(3^{3} \cdot 61\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1647} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1647,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;61\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;61\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + p T + p T^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T - T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T - 13 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T - 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 97 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 4 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 7 T + 49 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 13 T + 156 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 15 T + 183 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - T + 63 T^{2} - p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 3 T + 81 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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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

−19.0907352494, −18.8383521169, −18.2926779199, −17.7944677431, −17.214293814, −16.8349984064, −16.5023894861, −15.7738651958, −15.5197404554, −14.2454988041, −13.7479597857, −13.005675492, −12.520310078, −12.1460204264, −11.453523157, −10.3130886601, −10.0562302152, −9.63180123197, −9.03404968379, −7.94246116875, −6.88046184298, −6.40280474813, −5.65035023644, −4.64424977244, −3.66720215496, 0, 3.66720215496, 4.64424977244, 5.65035023644, 6.40280474813, 6.88046184298, 7.94246116875, 9.03404968379, 9.63180123197, 10.0562302152, 10.3130886601, 11.453523157, 12.1460204264, 12.520310078, 13.005675492, 13.7479597857, 14.2454988041, 15.5197404554, 15.7738651958, 16.5023894861, 16.8349984064, 17.214293814, 17.7944677431, 18.2926779199, 18.8383521169, 19.0907352494

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