Properties

Degree 4
Conductor 1327
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 + 4-s + 6·6-s − 4·7-s + 2·9-s + 11-s − 3·12-s − 3·13-s + 8·14-s + 16-s + 3·17-s − 4·18-s + 19-s + 12·21-s − 2·22-s − 23-s + 4·25-s + 6·26-s + 6·27-s − 4·28-s − 4·29-s − 13·31-s + 2·32-s − 3·33-s − 6·34-s + 2·36-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 1/2·4-s + 2.44·6-s − 1.51·7-s + 2/3·9-s + 0.301·11-s − 0.866·12-s − 0.832·13-s + 2.13·14-s + 1/4·16-s + 0.727·17-s − 0.942·18-s + 0.229·19-s + 2.61·21-s − 0.426·22-s − 0.208·23-s + 4/5·25-s + 1.17·26-s + 1.15·27-s − 0.755·28-s − 0.742·29-s − 2.33·31-s + 0.353·32-s − 0.522·33-s − 1.02·34-s + 1/3·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1327 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1327 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1327\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1327} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1327,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 1327$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 1327$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad1327$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 49 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 13 T + 101 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T - 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 42 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T - 114 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$D_{4}$ \( 1 + 8 T + 112 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 7 T + 77 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 3 T + 87 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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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

−19.4568132455, −19.1396760045, −18.5121394195, −17.9987659049, −17.7176505659, −16.9055272626, −16.7191035409, −16.4623848766, −15.7915242087, −14.7936348975, −14.3601336416, −13.1878424064, −12.6744462027, −12.1324685551, −11.4941021799, −10.9276137669, −10.0452144323, −9.79090990509, −9.09708784793, −8.38044268073, −7.21999681955, −6.63913002478, −5.76373374304, −5.17500850263, −3.40287607771, 0, 3.40287607771, 5.17500850263, 5.76373374304, 6.63913002478, 7.21999681955, 8.38044268073, 9.09708784793, 9.79090990509, 10.0452144323, 10.9276137669, 11.4941021799, 12.1324685551, 12.6744462027, 13.1878424064, 14.3601336416, 14.7936348975, 15.7915242087, 16.4623848766, 16.7191035409, 16.9055272626, 17.7176505659, 17.9987659049, 18.5121394195, 19.1396760045, 19.4568132455

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