Properties

Degree 4
Conductor $ 2^{5} \cdot 7^{2} \cdot 17^{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-s − 4·3-s + 4-s − 4·6-s − 4·7-s + 8-s + 6·9-s − 4·12-s − 4·14-s + 16-s + 6·18-s − 8·19-s + 16·21-s − 4·24-s − 10·25-s + 4·27-s − 4·28-s − 8·31-s + 32-s + 6·36-s − 8·37-s − 8·38-s + 16·42-s − 4·48-s + 9·49-s − 10·50-s − 12·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.63·6-s − 1.51·7-s + 0.353·8-s + 2·9-s − 1.15·12-s − 1.06·14-s + 1/4·16-s + 1.41·18-s − 1.83·19-s + 3.49·21-s − 0.816·24-s − 2·25-s + 0.769·27-s − 0.755·28-s − 1.43·31-s + 0.176·32-s + 36-s − 1.31·37-s − 1.29·38-s + 2.46·42-s − 0.577·48-s + 9/7·49-s − 1.41·50-s − 1.64·53-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 453152 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 453152 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(453152\)    =    \(2^{5} \cdot 7^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{453152} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 453152,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−8.119088937246101621284565887241, −7.30031963309496887656675542791, −6.74210935887901522750313637764, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −5.95059894955923738552113521496, −5.39319404309060655571342210758, −5.02122270959819660640979024866, −4.25383659493365434358528666274, −3.90229547122613769308947697962, −3.23512855113160871156144496989, −2.47165703864793927283611626754, −1.59349369718746011614691670371, 0, 0, 1.59349369718746011614691670371, 2.47165703864793927283611626754, 3.23512855113160871156144496989, 3.90229547122613769308947697962, 4.25383659493365434358528666274, 5.02122270959819660640979024866, 5.39319404309060655571342210758, 5.95059894955923738552113521496, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 6.74210935887901522750313637764, 7.30031963309496887656675542791, 8.119088937246101621284565887241

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