Properties

Degree 4
Conductor $ 2^{6} \cdot 61^{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-s − 4-s + 2·7-s + 3·8-s − 2·9-s − 2·14-s − 16-s + 8·17-s + 2·18-s − 18·23-s − 25-s − 2·28-s − 5·32-s − 8·34-s + 2·36-s + 10·41-s + 18·46-s + 8·47-s − 11·49-s + 50-s + 6·56-s − 4·63-s + 7·64-s − 8·68-s − 16·71-s − 6·72-s − 22·73-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s − 2/3·9-s − 0.534·14-s − 1/4·16-s + 1.94·17-s + 0.471·18-s − 3.75·23-s − 1/5·25-s − 0.377·28-s − 0.883·32-s − 1.37·34-s + 1/3·36-s + 1.56·41-s + 2.65·46-s + 1.16·47-s − 1.57·49-s + 0.141·50-s + 0.801·56-s − 0.503·63-s + 7/8·64-s − 0.970·68-s − 1.89·71-s − 0.707·72-s − 2.57·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 238144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 238144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(238144\)    =    \(2^{6} \cdot 61^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{238144} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 238144,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;61\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;61\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T + p T^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.569165758261804870177868073898, −8.287570693733701613432166498970, −7.85884968360616883145053235731, −7.66039763333771123227596439320, −7.12251875143434015278201977633, −6.08852912831720735332378674864, −5.72640884385550487126915198965, −5.60157612088078313975836309422, −4.59002554505366699590804892989, −4.25666915587159511841100490901, −3.69588602176779207669631056609, −2.89089326170447943989229722201, −1.96512649323539838938700359285, −1.31708422340871154028668296964, 0, 1.31708422340871154028668296964, 1.96512649323539838938700359285, 2.89089326170447943989229722201, 3.69588602176779207669631056609, 4.25666915587159511841100490901, 4.59002554505366699590804892989, 5.60157612088078313975836309422, 5.72640884385550487126915198965, 6.08852912831720735332378674864, 7.12251875143434015278201977633, 7.66039763333771123227596439320, 7.85884968360616883145053235731, 8.287570693733701613432166498970, 8.569165758261804870177868073898

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