Properties

Degree 4
Conductor $ 2^{4} \cdot 11^{2} \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·5-s − 5·9-s − 2·13-s + 4·16-s − 8·17-s + 4·20-s − 7·25-s − 4·29-s + 10·36-s − 22·37-s + 20·41-s + 10·45-s − 10·49-s + 4·52-s + 4·53-s − 4·61-s − 8·64-s + 4·65-s + 16·68-s − 32·73-s − 8·80-s + 16·81-s + 16·85-s − 14·89-s − 26·97-s + 14·100-s + ⋯
L(s)  = 1  − 4-s − 0.894·5-s − 5/3·9-s − 0.554·13-s + 16-s − 1.94·17-s + 0.894·20-s − 7/5·25-s − 0.742·29-s + 5/3·36-s − 3.61·37-s + 3.12·41-s + 1.49·45-s − 1.42·49-s + 0.554·52-s + 0.549·53-s − 0.512·61-s − 64-s + 0.496·65-s + 1.94·68-s − 3.74·73-s − 0.894·80-s + 16/9·81-s + 1.73·85-s − 1.48·89-s − 2.63·97-s + 7/5·100-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(327184\)    =    \(2^{4} \cdot 11^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{327184} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 327184,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
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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

−8.569451247588636860049749617820, −7.970223403211217946437728928671, −7.31285386083445627995852926185, −7.21661224338768342466420756206, −6.25972105104659001379078642019, −5.82135849339387357739621546932, −5.48444750282122420550662939059, −4.75768041533452306802121891710, −4.37350493013620998114647253846, −3.82887621952697788089432897883, −3.29304594675951575785139026233, −2.60654681549459182474671243206, −1.81431523402274649168122929883, 0, 0, 1.81431523402274649168122929883, 2.60654681549459182474671243206, 3.29304594675951575785139026233, 3.82887621952697788089432897883, 4.37350493013620998114647253846, 4.75768041533452306802121891710, 5.48444750282122420550662939059, 5.82135849339387357739621546932, 6.25972105104659001379078642019, 7.21661224338768342466420756206, 7.31285386083445627995852926185, 7.970223403211217946437728928671, 8.569451247588636860049749617820

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