Properties

Degree 4
Conductor $ 2^{18} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 4·9-s + 12·13-s + 2·25-s + 4·29-s + 12·37-s + 12·41-s − 16·45-s − 6·49-s − 4·53-s + 12·61-s + 48·65-s − 24·73-s + 7·81-s − 24·89-s − 16·97-s − 20·101-s − 12·109-s + 12·113-s − 48·117-s − 4·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + ⋯
L(s)  = 1  + 1.78·5-s − 4/3·9-s + 3.32·13-s + 2/5·25-s + 0.742·29-s + 1.97·37-s + 1.87·41-s − 2.38·45-s − 6/7·49-s − 0.549·53-s + 1.53·61-s + 5.95·65-s − 2.80·73-s + 7/9·81-s − 2.54·89-s − 1.62·97-s − 1.99·101-s − 1.14·109-s + 1.12·113-s − 4.43·117-s − 0.363·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(262144\)    =    \(2^{18}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{262144} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 262144,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.720951236$
$L(\frac12)$  $\approx$  $2.720951236$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 2$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
good3$V_4$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$V_4$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$V_4$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$V_4$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$V_4$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$V_4$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$V_4$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$V_4$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$V_4$ \( 1 - 28 T^{2} + p^{2} T^{4} \)
71$V_4$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
79$V_4$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 + 148 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−9.036800116186650427672448385071, −8.504509402954959248433237085570, −8.136209404913370480790908559261, −7.71500601574158399637037578135, −6.61387298753804977066819083208, −6.39786281520133230964526164360, −5.98859826938756647506654502941, −5.59569989859156343397361984183, −5.47761574613046253906233273467, −4.17878088743430670818200841248, −4.03141419798265746295869587755, −2.93642251168937868637046890158, −2.73263639610669961746742872748, −1.68277690453840397387089625321, −1.12736751813594213332646162785, 1.12736751813594213332646162785, 1.68277690453840397387089625321, 2.73263639610669961746742872748, 2.93642251168937868637046890158, 4.03141419798265746295869587755, 4.17878088743430670818200841248, 5.47761574613046253906233273467, 5.59569989859156343397361984183, 5.98859826938756647506654502941, 6.39786281520133230964526164360, 6.61387298753804977066819083208, 7.71500601574158399637037578135, 8.136209404913370480790908559261, 8.504509402954959248433237085570, 9.036800116186650427672448385071

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