Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{3} \cdot 11^{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  + 4-s − 5-s − 5·9-s + 2·11-s + 16-s − 14·19-s − 20-s + 25-s − 6·29-s − 14·31-s − 5·36-s + 12·41-s + 2·44-s + 5·45-s + 11·49-s − 2·55-s − 12·59-s − 2·61-s + 64-s + 6·71-s − 14·76-s − 20·79-s − 80-s + 16·81-s + 18·89-s + 14·95-s − 10·99-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.447·5-s − 5/3·9-s + 0.603·11-s + 1/4·16-s − 3.21·19-s − 0.223·20-s + 1/5·25-s − 1.11·29-s − 2.51·31-s − 5/6·36-s + 1.87·41-s + 0.301·44-s + 0.745·45-s + 11/7·49-s − 0.269·55-s − 1.56·59-s − 0.256·61-s + 1/8·64-s + 0.712·71-s − 1.60·76-s − 2.25·79-s − 0.111·80-s + 16/9·81-s + 1.90·89-s + 1.43·95-s − 1.00·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 60500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 60500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(60500\)    =    \(2^{2} \cdot 5^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{60500} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 60500,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( 1 + T \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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

−9.515625676781913720459205161908, −8.996361573480265218453634288731, −8.766840852787028026444155939270, −8.285662434899332908926558240119, −7.56981196469027501153188760806, −7.20457953717496815600620253355, −6.43810722624715010093850437821, −5.90185767008403089888905192216, −5.73876439085829035805370640147, −4.68830133406592764908928362464, −4.01173135747517138761270457828, −3.51902803001209100282256024354, −2.52121766859740692142668949704, −1.97170842587956421114470880781, 0, 1.97170842587956421114470880781, 2.52121766859740692142668949704, 3.51902803001209100282256024354, 4.01173135747517138761270457828, 4.68830133406592764908928362464, 5.73876439085829035805370640147, 5.90185767008403089888905192216, 6.43810722624715010093850437821, 7.20457953717496815600620253355, 7.56981196469027501153188760806, 8.285662434899332908926558240119, 8.766840852787028026444155939270, 8.996361573480265218453634288731, 9.515625676781913720459205161908

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