Properties

Degree 4
Conductor $ 2^{4} \cdot 7^{2} \cdot 11^{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 − 4·5-s − 3·8-s − 2·9-s − 4·10-s + 8·13-s − 16-s + 8·17-s − 2·18-s + 4·20-s + 2·25-s + 8·26-s − 12·29-s + 5·32-s + 8·34-s + 2·36-s − 12·37-s + 12·40-s + 8·41-s + 8·45-s + 49-s + 2·50-s − 8·52-s − 12·53-s − 12·58-s + 7·64-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s − 2/3·9-s − 1.26·10-s + 2.21·13-s − 1/4·16-s + 1.94·17-s − 0.471·18-s + 0.894·20-s + 2/5·25-s + 1.56·26-s − 2.22·29-s + 0.883·32-s + 1.37·34-s + 1/3·36-s − 1.97·37-s + 1.89·40-s + 1.24·41-s + 1.19·45-s + 1/7·49-s + 0.282·50-s − 1.10·52-s − 1.64·53-s − 1.57·58-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(94864\)    =    \(2^{4} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{94864} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 94864,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.275951467887346935408038775382, −8.678915900547504955372461266927, −8.401164460248213891696224277574, −7.82867506007281568614416512666, −7.60485396777068502870784654080, −6.83793327932833057571081045155, −5.96443978110022033983848919969, −5.70015900664701843854698011684, −5.30138461732919402727155679326, −4.11323776412142174518703250626, −4.07930883034575373640745113861, −3.26984949414789173092793201853, −3.26888344602981049454952325554, −1.43444095024382319596213609185, 0, 1.43444095024382319596213609185, 3.26888344602981049454952325554, 3.26984949414789173092793201853, 4.07930883034575373640745113861, 4.11323776412142174518703250626, 5.30138461732919402727155679326, 5.70015900664701843854698011684, 5.96443978110022033983848919969, 6.83793327932833057571081045155, 7.60485396777068502870784654080, 7.82867506007281568614416512666, 8.401164460248213891696224277574, 8.678915900547504955372461266927, 9.275951467887346935408038775382

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