Properties

Degree 4
Conductor $ 3^{2} \cdot 17^{3} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 9-s − 2·13-s + 12·16-s − 17-s − 2·19-s − 25-s − 4·36-s − 14·43-s − 12·47-s + 2·49-s + 8·52-s − 12·53-s + 12·59-s − 32·64-s − 8·67-s + 4·68-s + 8·76-s + 81-s − 12·83-s + 4·100-s + 10·103-s − 2·117-s − 13·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 2·4-s + 1/3·9-s − 0.554·13-s + 3·16-s − 0.242·17-s − 0.458·19-s − 1/5·25-s − 2/3·36-s − 2.13·43-s − 1.75·47-s + 2/7·49-s + 1.10·52-s − 1.64·53-s + 1.56·59-s − 4·64-s − 0.977·67-s + 0.485·68-s + 0.917·76-s + 1/9·81-s − 1.31·83-s + 2/5·100-s + 0.985·103-s − 0.184·117-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 44217 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 44217 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(44217\)    =    \(3^{2} \cdot 17^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{44217} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 44217,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.721668236319765682540445072382, −9.642727340101070443548067847258, −8.747358907070908179484393186464, −8.618966621189813626434174913816, −7.960547248828429430124687340941, −7.53511853023675535658862215085, −6.66463575826783700250676263362, −6.12714514378066180257024905951, −5.14306131647545403565477997005, −5.04692212052117448633199558052, −4.31112644939450006748261353184, −3.76588854630918978975078187598, −3.00632648437536992753290178800, −1.57155610960796422893905590944, 0, 1.57155610960796422893905590944, 3.00632648437536992753290178800, 3.76588854630918978975078187598, 4.31112644939450006748261353184, 5.04692212052117448633199558052, 5.14306131647545403565477997005, 6.12714514378066180257024905951, 6.66463575826783700250676263362, 7.53511853023675535658862215085, 7.960547248828429430124687340941, 8.618966621189813626434174913816, 8.747358907070908179484393186464, 9.642727340101070443548067847258, 9.721668236319765682540445072382

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