Properties

Degree 4
Conductor $ 2^{7} \cdot 3^{2} \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 2·12-s + 16-s + 18-s − 8·19-s − 2·24-s − 10·25-s + 4·27-s + 32-s + 36-s − 8·38-s + 16·43-s − 2·48-s + 2·49-s − 10·50-s − 12·53-s + 4·54-s + 16·57-s + 64-s + 16·67-s + 72-s + 4·73-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 0.235·18-s − 1.83·19-s − 0.408·24-s − 2·25-s + 0.769·27-s + 0.176·32-s + 1/6·36-s − 1.29·38-s + 2.43·43-s − 0.288·48-s + 2/7·49-s − 1.41·50-s − 1.64·53-s + 0.544·54-s + 2.11·57-s + 1/8·64-s + 1.95·67-s + 0.117·72-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(332928\)    =    \(2^{7} \cdot 3^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{332928} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 332928,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.375183600$
$L(\frac12)$  $\approx$  $1.375183600$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 - T \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−8.714241774370998291971873352051, −8.219888797179871881100664672892, −7.71319623971707540470742927921, −7.29484938101166108994999958784, −6.65330731261113455462227374582, −6.15777205658123834208045946258, −5.99911855171913780844071238816, −5.55324695398511660179242639164, −4.78136048868217964603743789674, −4.53177870298203396965614639892, −3.90229547122613769308947697962, −3.37176613840163753335045610201, −2.35027024394452387135343152508, −1.96477727170631410018279771361, −0.62289611347536456250264873990, 0.62289611347536456250264873990, 1.96477727170631410018279771361, 2.35027024394452387135343152508, 3.37176613840163753335045610201, 3.90229547122613769308947697962, 4.53177870298203396965614639892, 4.78136048868217964603743789674, 5.55324695398511660179242639164, 5.99911855171913780844071238816, 6.15777205658123834208045946258, 6.65330731261113455462227374582, 7.29484938101166108994999958784, 7.71319623971707540470742927921, 8.219888797179871881100664672892, 8.714241774370998291971873352051

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