Properties

Degree 4
Conductor $ 2^{2} \cdot 19^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 6-s − 2·7-s + 8-s + 3·9-s − 12·11-s + 5·13-s + 2·14-s − 16-s − 3·17-s − 3·18-s − 2·21-s + 12·22-s − 3·23-s + 24-s + 5·25-s − 5·26-s + 8·27-s + 9·29-s + 8·31-s − 12·33-s + 3·34-s − 4·37-s + 5·39-s + 2·42-s − 8·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 9-s − 3.61·11-s + 1.38·13-s + 0.534·14-s − 1/4·16-s − 0.727·17-s − 0.707·18-s − 0.436·21-s + 2.55·22-s − 0.625·23-s + 0.204·24-s + 25-s − 0.980·26-s + 1.53·27-s + 1.67·29-s + 1.43·31-s − 2.08·33-s + 0.514·34-s − 0.657·37-s + 0.800·39-s + 0.308·42-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(521284\)    =    \(2^{2} \cdot 19^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{722} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 521284,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.913461\)
\(L(\frac12)\)  \(\approx\)  \(0.913461\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;19\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + T^{2} \)
19 \( 1 \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.53077602142962907183313860133, −10.22101133251579735860213114071, −9.796749249224507096433177692171, −9.546760410176431520553977692956, −8.537926150676000485622522402607, −8.419819136123545691116628041944, −8.239521637892575084687539446303, −7.942568995500605341179616020238, −7.11712316196877396218758258705, −6.69230721480135968216691032129, −6.57537934232554710442037647549, −5.50040037976405017813878689585, −5.36661437045520252495305862638, −4.47195322090585336419526387351, −4.45902863949864490936020962716, −3.28887938852658383790051091168, −2.88917259317035582843155482657, −2.56942305234563306728464640756, −1.61558824482515643800911368613, −0.54032630809861117031678362844, 0.54032630809861117031678362844, 1.61558824482515643800911368613, 2.56942305234563306728464640756, 2.88917259317035582843155482657, 3.28887938852658383790051091168, 4.45902863949864490936020962716, 4.47195322090585336419526387351, 5.36661437045520252495305862638, 5.50040037976405017813878689585, 6.57537934232554710442037647549, 6.69230721480135968216691032129, 7.11712316196877396218758258705, 7.942568995500605341179616020238, 8.239521637892575084687539446303, 8.419819136123545691116628041944, 8.537926150676000485622522402607, 9.546760410176431520553977692956, 9.796749249224507096433177692171, 10.22101133251579735860213114071, 10.53077602142962907183313860133

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