Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{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  − 4-s + 16-s − 4·19-s − 6·29-s − 2·31-s − 6·41-s − 49-s − 6·59-s − 2·61-s − 64-s + 4·76-s − 16·79-s − 12·89-s − 36·101-s − 4·109-s + 6·116-s − 22·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/4·16-s − 0.917·19-s − 1.11·29-s − 0.359·31-s − 0.937·41-s − 1/7·49-s − 0.781·59-s − 0.256·61-s − 1/8·64-s + 0.458·76-s − 1.80·79-s − 1.27·89-s − 3.58·101-s − 0.383·109-s + 0.557·116-s − 2·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.468·164-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9922500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3013376667$
$L(\frac12)$  $\approx$  $0.3013376667$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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

−8.926240425076139529861135002547, −8.365065907294969951319741632313, −8.332642460642684211519171969751, −7.67054796206946527669475760269, −7.53796247359455911496786008721, −6.88069824419769952242560770924, −6.73372141171708547299939532131, −6.20666347786653201093634544482, −5.79772447578655773769943597364, −5.40919411310355706303595223420, −5.09722438654297425079593523972, −4.56497589414483073582353861239, −4.17885976969187413867762469923, −3.79512642451380361793118861554, −3.44089148561824895520842404339, −2.68174709079835084957881115169, −2.49738694601324945227769951780, −1.50070957336960335228174065644, −1.45381212515555590144940713589, −0.17003627585707894682212660472, 0.17003627585707894682212660472, 1.45381212515555590144940713589, 1.50070957336960335228174065644, 2.49738694601324945227769951780, 2.68174709079835084957881115169, 3.44089148561824895520842404339, 3.79512642451380361793118861554, 4.17885976969187413867762469923, 4.56497589414483073582353861239, 5.09722438654297425079593523972, 5.40919411310355706303595223420, 5.79772447578655773769943597364, 6.20666347786653201093634544482, 6.73372141171708547299939532131, 6.88069824419769952242560770924, 7.53796247359455911496786008721, 7.67054796206946527669475760269, 8.332642460642684211519171969751, 8.365065907294969951319741632313, 8.926240425076139529861135002547

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