Properties

Degree 4
Conductor $ 139 \cdot 449 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s − 5·5-s + 6·6-s − 5·7-s + 4·8-s + 2·9-s + 10·10-s − 11-s − 9·13-s + 10·14-s + 15·15-s − 4·16-s − 3·17-s − 4·18-s − 11·19-s + 15·21-s + 2·22-s + 6·23-s − 12·24-s + 10·25-s + 18·26-s + 6·27-s − 30·30-s + 3·33-s + 6·34-s + 25·35-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s − 2.23·5-s + 2.44·6-s − 1.88·7-s + 1.41·8-s + 2/3·9-s + 3.16·10-s − 0.301·11-s − 2.49·13-s + 2.67·14-s + 3.87·15-s − 16-s − 0.727·17-s − 0.942·18-s − 2.52·19-s + 3.27·21-s + 0.426·22-s + 1.25·23-s − 2.44·24-s + 2·25-s + 3.53·26-s + 1.15·27-s − 5.47·30-s + 0.522·33-s + 1.02·34-s + 4.22·35-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 62411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 62411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(62411\)    =    \(139 \cdot 449\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{62411} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 62411,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{139,\;449\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{139,\;449\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad139$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 19 T + p T^{2} ) \)
449$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 5 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$D_{4}$ \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$D_{4}$ \( 1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 11 T + 80 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 21 T + 240 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 5 T + 190 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
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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

−15.3422516609, −15.1174564803, −14.538598798, −13.6936175081, −13.1013058984, −12.6268817914, −12.299234141, −12.0831248173, −11.5148374673, −11.0473329143, −10.4889514554, −10.2265020023, −9.85694791456, −9.00659510787, −8.65922173865, −8.43599013863, −7.43030174813, −7.26523399038, −6.71447039636, −6.24552502828, −5.22545540369, −4.80616274233, −4.28633774146, −3.53074242944, −2.60688912206, 0, 0, 0, 2.60688912206, 3.53074242944, 4.28633774146, 4.80616274233, 5.22545540369, 6.24552502828, 6.71447039636, 7.26523399038, 7.43030174813, 8.43599013863, 8.65922173865, 9.00659510787, 9.85694791456, 10.2265020023, 10.4889514554, 11.0473329143, 11.5148374673, 12.0831248173, 12.299234141, 12.6268817914, 13.1013058984, 13.6936175081, 14.538598798, 15.1174564803, 15.3422516609

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