Properties

Degree 4
Conductor $ 3 \cdot 359 $
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 − 3·5-s + 6·6-s − 7-s + 4·9-s + 6·10-s + 2·11-s − 3·12-s − 2·13-s + 2·14-s + 9·15-s + 16-s − 3·17-s − 8·18-s + 19-s − 3·20-s + 3·21-s − 4·22-s − 4·23-s + 25-s + 4·26-s − 28-s − 6·29-s − 18·30-s − 31-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s − 0.377·7-s + 4/3·9-s + 1.89·10-s + 0.603·11-s − 0.866·12-s − 0.554·13-s + 0.534·14-s + 2.32·15-s + 1/4·16-s − 0.727·17-s − 1.88·18-s + 0.229·19-s − 0.670·20-s + 0.654·21-s − 0.852·22-s − 0.834·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 1.11·29-s − 3.28·30-s − 0.179·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1077\)    =    \(3 \cdot 359\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1077} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1077,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;359\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;359\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
359$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 8 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$V_4$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
11$V_4$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$D_{4}$ \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$V_4$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 114 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 13 T + 174 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 154 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$V_4$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−19.9792106538, −19.3768544424, −18.9087295598, −18.3724983167, −17.8207373097, −17.4724129722, −16.9514753465, −16.4066464564, −16.0299464838, −15.3482491336, −14.7611109666, −13.753632319, −12.80787039, −12.0522309383, −11.8676959823, −11.135273889, −10.7006019347, −9.70008827668, −9.3380254628, −8.32296493864, −7.71918412015, −6.80397864306, −6.08591447925, −4.96180986609, −3.87679222455, 0, 3.87679222455, 4.96180986609, 6.08591447925, 6.80397864306, 7.71918412015, 8.32296493864, 9.3380254628, 9.70008827668, 10.7006019347, 11.135273889, 11.8676959823, 12.0522309383, 12.80787039, 13.753632319, 14.7611109666, 15.3482491336, 16.0299464838, 16.4066464564, 16.9514753465, 17.4724129722, 17.8207373097, 18.3724983167, 18.9087295598, 19.3768544424, 19.9792106538

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