Properties

Degree 4
Conductor $ 2^{3} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 2·4-s − 2·5-s + 8·6-s − 4·7-s + 7·9-s + 4·10-s − 8·12-s + 4·13-s + 8·14-s + 8·15-s − 4·16-s − 8·17-s − 14·18-s + 4·19-s − 4·20-s + 16·21-s − 3·25-s − 8·26-s − 4·27-s − 8·28-s − 8·29-s − 16·30-s + 8·32-s + 16·34-s + 8·35-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 4-s − 0.894·5-s + 3.26·6-s − 1.51·7-s + 7/3·9-s + 1.26·10-s − 2.30·12-s + 1.10·13-s + 2.13·14-s + 2.06·15-s − 16-s − 1.94·17-s − 3.29·18-s + 0.917·19-s − 0.894·20-s + 3.49·21-s − 3/5·25-s − 1.56·26-s − 0.769·27-s − 1.51·28-s − 1.48·29-s − 2.92·30-s + 1.41·32-s + 2.74·34-s + 1.35·35-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(968\)    =    \(2^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{968} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 968,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−19.3981440797, −19.1857249719, −18.4484037676, −17.9414335735, −17.6304228292, −17.0336103204, −16.4150110915, −16.0208509362, −15.9140726033, −15.1389643255, −13.5686390571, −13.1644289392, −12.3870421342, −11.4512586103, −11.3646694668, −10.8920468072, −10.0355090972, −9.40167023369, −8.60353961929, −7.57356519541, −6.61017530637, −6.36261389471, −5.26718062024, −3.92131200413, 0, 3.92131200413, 5.26718062024, 6.36261389471, 6.61017530637, 7.57356519541, 8.60353961929, 9.40167023369, 10.0355090972, 10.8920468072, 11.3646694668, 11.4512586103, 12.3870421342, 13.1644289392, 13.5686390571, 15.1389643255, 15.9140726033, 16.0208509362, 16.4150110915, 17.0336103204, 17.6304228292, 17.9414335735, 18.4484037676, 19.1857249719, 19.3981440797

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