Properties

Degree 4
Conductor 1499
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 − 2·3-s + 4-s − 3·5-s + 4·6-s − 7-s + 2·9-s + 6·10-s − 7·11-s − 2·12-s + 4·13-s + 2·14-s + 6·15-s + 16-s + 4·17-s − 4·18-s − 3·19-s − 3·20-s + 2·21-s + 14·22-s − 3·23-s + 2·25-s − 8·26-s − 6·27-s − 28-s + 4·29-s − 12·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s + 1.63·6-s − 0.377·7-s + 2/3·9-s + 1.89·10-s − 2.11·11-s − 0.577·12-s + 1.10·13-s + 0.534·14-s + 1.54·15-s + 1/4·16-s + 0.970·17-s − 0.942·18-s − 0.688·19-s − 0.670·20-s + 0.436·21-s + 2.98·22-s − 0.625·23-s + 2/5·25-s − 1.56·26-s − 1.15·27-s − 0.188·28-s + 0.742·29-s − 2.19·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1499 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1499 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1499\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1499} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1499,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 1499$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 1499$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad1499$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 36 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 7 T + 3 p T^{2} + 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T - 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$D_{4}$ \( 1 + 5 T + 83 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 166 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 3 T - 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 175 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−19.2652029943, −18.9056954065, −18.404492076, −18.0020772672, −17.7086914145, −16.736421355, −16.5713789365, −15.763462506, −15.6186952622, −15.050884539, −13.8017601704, −13.2385638377, −12.424120959, −12.0816511382, −11.1837819079, −10.8696964455, −10.1301173928, −9.69388579168, −8.5592362584, −7.98041829618, −7.78090172012, −6.51655485872, −5.7176519566, −4.7263489311, −3.40759643562, 0, 3.40759643562, 4.7263489311, 5.7176519566, 6.51655485872, 7.78090172012, 7.98041829618, 8.5592362584, 9.69388579168, 10.1301173928, 10.8696964455, 11.1837819079, 12.0816511382, 12.424120959, 13.2385638377, 13.8017601704, 15.050884539, 15.6186952622, 15.763462506, 16.5713789365, 16.736421355, 17.7086914145, 18.0020772672, 18.404492076, 18.9056954065, 19.2652029943

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