Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{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  + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 2·9-s + 12·11-s − 8·14-s + 5·16-s − 4·18-s + 24·22-s − 10·25-s − 12·28-s + 6·32-s − 6·36-s − 8·37-s + 16·43-s + 36·44-s + 9·49-s − 20·50-s − 12·53-s − 16·56-s + 8·63-s + 7·64-s + 16·67-s − 8·72-s − 16·74-s − 48·77-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 2/3·9-s + 3.61·11-s − 2.13·14-s + 5/4·16-s − 0.942·18-s + 5.11·22-s − 2·25-s − 2.26·28-s + 1.06·32-s − 36-s − 1.31·37-s + 2.43·43-s + 5.42·44-s + 9/7·49-s − 2.82·50-s − 1.64·53-s − 2.13·56-s + 1.00·63-s + 7/8·64-s + 1.95·67-s − 0.942·72-s − 1.85·74-s − 5.47·77-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(56644\)    =    \(2^{2} \cdot 7^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{56644} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 56644,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.167608159$
$L(\frac12)$  $\approx$  $3.167608159$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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

−9.795849732674491714602252272054, −9.435449454754526833910951528272, −9.286071988264369821451329617999, −8.550482821819457584609282080376, −7.75961744967946030738916282437, −6.98368217520104403156128027372, −6.65330731261113455462227374582, −6.16055468076817207109148001002, −5.99911855171913780844071238816, −5.16488039277883802975012518677, −4.04100934533934329070442787144, −3.90229547122613769308947697962, −3.50747685387131529534531471772, −2.52652883309992049334898398402, −1.44781063014612234378860428910, 1.44781063014612234378860428910, 2.52652883309992049334898398402, 3.50747685387131529534531471772, 3.90229547122613769308947697962, 4.04100934533934329070442787144, 5.16488039277883802975012518677, 5.99911855171913780844071238816, 6.16055468076817207109148001002, 6.65330731261113455462227374582, 6.98368217520104403156128027372, 7.75961744967946030738916282437, 8.550482821819457584609282080376, 9.286071988264369821451329617999, 9.435449454754526833910951528272, 9.795849732674491714602252272054

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