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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 8·11-s + 16-s + 8·19-s − 4·29-s + 12·41-s − 8·44-s − 49-s + 8·59-s + 12·61-s − 64-s − 16·71-s − 8·76-s − 12·89-s + 4·101-s + 4·109-s + 4·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 12·164-s + ⋯
L(s)  = 1  − 1/2·4-s + 2.41·11-s + 1/4·16-s + 1.83·19-s − 0.742·29-s + 1.87·41-s − 1.20·44-s − 1/7·49-s + 1.04·59-s + 1.53·61-s − 1/8·64-s − 1.89·71-s − 0.917·76-s − 1.27·89-s + 0.398·101-s + 0.383·109-s + 0.371·116-s + 2.36·121-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 − 0.937·164-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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.220996007$
$L(\frac12)$  $\approx$  $3.220996007$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.825662352621021287999239670911, −8.575688041178342675376580720894, −8.310515878353982249337371850513, −7.58708883784009702265886652514, −7.24352896766114055981756508196, −7.24146641322850673576644518077, −6.59696200645346197085099445930, −6.18226285667188940045780795352, −5.88122106670547835414241782326, −5.44936849187633603931885850179, −5.10187447989534779390674500199, −4.34336573609305764327316516822, −4.32895092309883762214038376186, −3.65610088890848707296211559053, −3.54646280687156728472762582249, −2.91013921001833239425327250775, −2.31982376803787028489023508474, −1.56770016999091544996877878190, −1.18213511157640488656663341229, −0.65489759606315591206840315201, 0.65489759606315591206840315201, 1.18213511157640488656663341229, 1.56770016999091544996877878190, 2.31982376803787028489023508474, 2.91013921001833239425327250775, 3.54646280687156728472762582249, 3.65610088890848707296211559053, 4.32895092309883762214038376186, 4.34336573609305764327316516822, 5.10187447989534779390674500199, 5.44936849187633603931885850179, 5.88122106670547835414241782326, 6.18226285667188940045780795352, 6.59696200645346197085099445930, 7.24146641322850673576644518077, 7.24352896766114055981756508196, 7.58708883784009702265886652514, 8.310515878353982249337371850513, 8.575688041178342675376580720894, 8.825662352621021287999239670911

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