Properties

Degree 4
Conductor $ 3 \cdot 83 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 4-s + 4·6-s − 7-s + 4·9-s + 11-s − 2·12-s + 2·14-s + 16-s − 17-s − 8·18-s − 6·19-s + 2·21-s − 2·22-s − 2·25-s − 5·27-s − 28-s + 3·29-s + 3·31-s + 2·32-s − 2·33-s + 2·34-s + 4·36-s − 5·37-s + 12·38-s + 4·41-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s − 0.377·7-s + 4/3·9-s + 0.301·11-s − 0.577·12-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 1.88·18-s − 1.37·19-s + 0.436·21-s − 0.426·22-s − 2/5·25-s − 0.962·27-s − 0.188·28-s + 0.557·29-s + 0.538·31-s + 0.353·32-s − 0.348·33-s + 0.342·34-s + 2/3·36-s − 0.821·37-s + 1.94·38-s + 0.624·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(249\)    =    \(3 \cdot 83\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{249} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 249,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.1315495070$
$L(\frac12)$  $\approx$  $0.1315495070$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;83\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;83\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T - 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 13 T + 106 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T + 48 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$D_{4}$ \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
97$D_{4}$ \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−19.958336202, −19.25325453, −18.8911406702, −18.3038693797, −17.5495656198, −17.3478671179, −16.7780994692, −15.9022638292, −15.5120887195, −14.4574259478, −13.4301254485, −12.5870403887, −12.0374010604, −11.0180573664, −10.412169202, −9.68468363219, −8.9532007232, −8.02996189451, −6.82879614383, −6.02680203091, −4.45419105482, 4.45419105482, 6.02680203091, 6.82879614383, 8.02996189451, 8.9532007232, 9.68468363219, 10.412169202, 11.0180573664, 12.0374010604, 12.5870403887, 13.4301254485, 14.4574259478, 15.5120887195, 15.9022638292, 16.7780994692, 17.3478671179, 17.5495656198, 18.3038693797, 18.8911406702, 19.25325453, 19.958336202

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