Properties

Degree 4
Conductor $ 3^{3} \cdot 11^{3} $
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  − 2·2-s − 3-s − 4-s + 2·6-s + 8·8-s + 9-s − 11-s + 12-s − 7·16-s + 4·17-s − 2·18-s + 2·22-s − 8·24-s − 6·25-s − 27-s + 12·29-s − 16·31-s − 14·32-s + 33-s − 8·34-s − 36-s + 12·37-s + 4·41-s + 44-s + 7·48-s + 2·49-s + 12·50-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s − 1/2·4-s + 0.816·6-s + 2.82·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 7/4·16-s + 0.970·17-s − 0.471·18-s + 0.426·22-s − 1.63·24-s − 6/5·25-s − 0.192·27-s + 2.22·29-s − 2.87·31-s − 2.47·32-s + 0.174·33-s − 1.37·34-s − 1/6·36-s + 1.97·37-s + 0.624·41-s + 0.150·44-s + 1.01·48-s + 2/7·49-s + 1.69·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35937 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35937 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(35937\)    =    \(3^{3} \cdot 11^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{35937} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 35937,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3278322794$
$L(\frac12)$  $\approx$  $0.3278322794$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( 1 + T \)
11$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.39134307397473772625111872930, −9.637990407765945223554197990480, −9.528487026476181059669700861222, −8.829662012954888204160857557729, −8.389268452519734683124359977272, −7.67376126887882242787787369865, −7.60731726518247812897599595849, −6.85553577597705614902881023761, −5.66464594246847012421851082372, −5.64913594526162848677189952595, −4.52395871965258729630923756023, −4.33065348032451946577135661956, −3.31119773648628104035349562784, −1.82001999803537272093437549398, −0.71891279628917143719551815370, 0.71891279628917143719551815370, 1.82001999803537272093437549398, 3.31119773648628104035349562784, 4.33065348032451946577135661956, 4.52395871965258729630923756023, 5.64913594526162848677189952595, 5.66464594246847012421851082372, 6.85553577597705614902881023761, 7.60731726518247812897599595849, 7.67376126887882242787787369865, 8.389268452519734683124359977272, 8.829662012954888204160857557729, 9.528487026476181059669700861222, 9.637990407765945223554197990480, 10.39134307397473772625111872930

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