Properties

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

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s − 2·9-s − 8·13-s + 5·16-s + 6·17-s + 4·18-s + 4·19-s − 10·25-s + 16·26-s − 6·32-s − 12·34-s − 6·36-s − 8·38-s + 16·43-s − 24·47-s + 49-s + 20·50-s − 24·52-s + 12·53-s − 12·59-s + 7·64-s − 8·67-s + 18·68-s + 8·72-s + 12·76-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2/3·9-s − 2.21·13-s + 5/4·16-s + 1.45·17-s + 0.942·18-s + 0.917·19-s − 2·25-s + 3.13·26-s − 1.06·32-s − 2.05·34-s − 36-s − 1.29·38-s + 2.43·43-s − 3.50·47-s + 1/7·49-s + 2.82·50-s − 3.32·52-s + 1.64·53-s − 1.56·59-s + 7/8·64-s − 0.977·67-s + 2.18·68-s + 0.942·72-s + 1.37·76-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  =  1
Selberg data  =  $(4,\ 56644,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
show more
show less
\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.765547119459919407856461234632, −9.440338822267021662634471861951, −8.812642335551285956694248042850, −8.090068662745416752784621377032, −7.61488245805577994602083893658, −7.57571100088867902110310233811, −6.87015167727116618005628901277, −6.04516833074826431165424798565, −5.57928681742950427486583645839, −5.05791232432713970996781243183, −4.04379426959865020166490254054, −3.04116088637864497579983367990, −2.57969057241224021562840012921, −1.54182063036964932495656311078, 0, 1.54182063036964932495656311078, 2.57969057241224021562840012921, 3.04116088637864497579983367990, 4.04379426959865020166490254054, 5.05791232432713970996781243183, 5.57928681742950427486583645839, 6.04516833074826431165424798565, 6.87015167727116618005628901277, 7.57571100088867902110310233811, 7.61488245805577994602083893658, 8.090068662745416752784621377032, 8.812642335551285956694248042850, 9.440338822267021662634471861951, 9.765547119459919407856461234632

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