Properties

Degree 4
Conductor $ 2^{3} \cdot 3^{2} \cdot 13 $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 4-s − 2·6-s − 4·7-s − 3·8-s + 3·9-s + 8·11-s + 2·12-s − 13-s − 4·14-s − 16-s + 4·17-s + 3·18-s − 4·19-s + 8·21-s + 8·22-s − 8·23-s + 6·24-s − 6·25-s − 26-s − 4·27-s + 4·28-s − 4·29-s + 12·31-s + 5·32-s − 16·33-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 1.51·7-s − 1.06·8-s + 9-s + 2.41·11-s + 0.577·12-s − 0.277·13-s − 1.06·14-s − 1/4·16-s + 0.970·17-s + 0.707·18-s − 0.917·19-s + 1.74·21-s + 1.70·22-s − 1.66·23-s + 1.22·24-s − 6/5·25-s − 0.196·26-s − 0.769·27-s + 0.755·28-s − 0.742·29-s + 2.15·31-s + 0.883·32-s − 2.78·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{936} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 936,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4456913213$
$L(\frac12)$  $\approx$  $0.4456913213$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−19.8934316645, −19.4604421777, −19.1551537949, −18.4282714936, −17.67075195, −17.2849148009, −16.6272610009, −16.4135833599, −15.4732516475, −14.9380436528, −14.1819882616, −13.6745032861, −12.9568449998, −12.3172568607, −11.8538765065, −11.5820474614, −9.96467191573, −9.95280234664, −9.19274055467, −8.09899069409, −6.54556545043, −6.42897107073, −5.6548678028, −4.25303028693, −3.71205447024, 3.71205447024, 4.25303028693, 5.6548678028, 6.42897107073, 6.54556545043, 8.09899069409, 9.19274055467, 9.95280234664, 9.96467191573, 11.5820474614, 11.8538765065, 12.3172568607, 12.9568449998, 13.6745032861, 14.1819882616, 14.9380436528, 15.4732516475, 16.4135833599, 16.6272610009, 17.2849148009, 17.67075195, 18.4282714936, 19.1551537949, 19.4604421777, 19.8934316645

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