Properties

Degree 4
Conductor $ 3^{4} \cdot 19 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·7-s − 2·13-s − 3·16-s + 3·19-s − 4·25-s + 2·28-s + 10·31-s + 4·37-s + 4·43-s − 2·49-s + 2·52-s − 8·61-s + 7·64-s − 2·67-s + 4·73-s − 3·76-s + 4·79-s + 4·91-s − 14·97-s + 4·100-s − 8·103-s − 14·109-s + 6·112-s + 14·121-s − 10·124-s + 127-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.755·7-s − 0.554·13-s − 3/4·16-s + 0.688·19-s − 4/5·25-s + 0.377·28-s + 1.79·31-s + 0.657·37-s + 0.609·43-s − 2/7·49-s + 0.277·52-s − 1.02·61-s + 7/8·64-s − 0.244·67-s + 0.468·73-s − 0.344·76-s + 0.450·79-s + 0.419·91-s − 1.42·97-s + 2/5·100-s − 0.788·103-s − 1.34·109-s + 0.566·112-s + 1.27·121-s − 0.898·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1539\)    =    \(3^{4} \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1539} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1539,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5221435606$
$L(\frac12)$  $\approx$  $0.5221435606$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3 \( 1 \)
19$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−13.68816362801267509972038229581, −13.15510058006465130171422176517, −12.40095477290838686937624430215, −11.89355091746708574088114751364, −11.20226839691820943158562643086, −10.34859675467964656496681254428, −9.600392817368737572042804110266, −9.384260671410315056207093168535, −8.389258775251883182185131053515, −7.66716851151729354484994997123, −6.78439873765208802194785716556, −6.03804535320455140917727072081, −4.98638130370676266639532417421, −4.09729391479811292149019045922, −2.78666419482573957815444693787, 2.78666419482573957815444693787, 4.09729391479811292149019045922, 4.98638130370676266639532417421, 6.03804535320455140917727072081, 6.78439873765208802194785716556, 7.66716851151729354484994997123, 8.389258775251883182185131053515, 9.384260671410315056207093168535, 9.600392817368737572042804110266, 10.34859675467964656496681254428, 11.20226839691820943158562643086, 11.89355091746708574088114751364, 12.40095477290838686937624430215, 13.15510058006465130171422176517, 13.68816362801267509972038229581

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