Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} $
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  − 2·2-s + 3·4-s + 5·7-s − 4·8-s − 5·9-s + 2·11-s − 10·14-s + 5·16-s + 10·18-s − 4·22-s − 12·23-s + 25-s + 15·28-s − 6·29-s − 6·32-s − 15·36-s − 14·37-s + 16·43-s + 6·44-s + 24·46-s + 18·49-s − 2·50-s − 6·53-s − 20·56-s + 12·58-s − 25·63-s + 7·64-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.88·7-s − 1.41·8-s − 5/3·9-s + 0.603·11-s − 2.67·14-s + 5/4·16-s + 2.35·18-s − 0.852·22-s − 2.50·23-s + 1/5·25-s + 2.83·28-s − 1.11·29-s − 1.06·32-s − 5/2·36-s − 2.30·37-s + 2.43·43-s + 0.904·44-s + 3.53·46-s + 18/7·49-s − 0.282·50-s − 0.824·53-s − 2.67·56-s + 1.57·58-s − 3.14·63-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(592900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{592900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 592900,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.8307506751$
$L(\frac12)$  $\approx$  $0.8307506751$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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

−8.285662434899332908926558240119, −8.283254682681969568742412941640, −7.56981196469027501153188760806, −7.44617088698970221673072177182, −6.75549642500640203917529221375, −6.07909319757312535714712792824, −5.73876439085829035805370640147, −5.44015915433775770833668591547, −4.72556975951210881700191573231, −4.01173135747517138761270457828, −3.53795176024353637912410403117, −2.64632677110737282548016864035, −1.97170842587956421114470880781, −1.74843844460655954658244071452, −0.57168783779040589515932752100, 0.57168783779040589515932752100, 1.74843844460655954658244071452, 1.97170842587956421114470880781, 2.64632677110737282548016864035, 3.53795176024353637912410403117, 4.01173135747517138761270457828, 4.72556975951210881700191573231, 5.44015915433775770833668591547, 5.73876439085829035805370640147, 6.07909319757312535714712792824, 6.75549642500640203917529221375, 7.44617088698970221673072177182, 7.56981196469027501153188760806, 8.283254682681969568742412941640, 8.285662434899332908926558240119

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