Properties

Degree 4
Conductor $ 3^{2} \cdot 31^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 4·4-s − 4·5-s + 6·6-s − 4·7-s − 3·8-s + 3·9-s + 12·10-s − 6·11-s − 8·12-s − 2·13-s + 12·14-s + 8·15-s + 3·16-s − 4·17-s − 9·18-s − 8·19-s − 16·20-s + 8·21-s + 18·22-s + 2·23-s + 6·24-s + 7·25-s + 6·26-s − 4·27-s − 16·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s − 1.78·5-s + 2.44·6-s − 1.51·7-s − 1.06·8-s + 9-s + 3.79·10-s − 1.80·11-s − 2.30·12-s − 0.554·13-s + 3.20·14-s + 2.06·15-s + 3/4·16-s − 0.970·17-s − 2.12·18-s − 1.83·19-s − 3.57·20-s + 1.74·21-s + 3.83·22-s + 0.417·23-s + 1.22·24-s + 7/5·25-s + 1.17·26-s − 0.769·27-s − 3.02·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(8649\)    =    \(3^{2} \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8649} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 8649,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;31\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + T )^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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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

−17.4689139414, −16.7023248629, −16.503898631, −15.9939202783, −15.6296819285, −15.2502658257, −14.7930293151, −13.2832504977, −13.112808071, −12.4013902912, −12.209584831, −11.3102701663, −10.8657136415, −10.5149981434, −10.0305388687, −9.44620632958, −8.78912796545, −8.09790154061, −7.90847101234, −6.96212014036, −6.7660766639, −5.76260028749, −4.76830701724, −3.93958617827, −2.71523764769, 0, 0, 2.71523764769, 3.93958617827, 4.76830701724, 5.76260028749, 6.7660766639, 6.96212014036, 7.90847101234, 8.09790154061, 8.78912796545, 9.44620632958, 10.0305388687, 10.5149981434, 10.8657136415, 11.3102701663, 12.209584831, 12.4013902912, 13.112808071, 13.2832504977, 14.7930293151, 15.2502658257, 15.6296819285, 15.9939202783, 16.503898631, 16.7023248629, 17.4689139414

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