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 + 16-s + 8·19-s − 12·29-s − 8·31-s − 12·41-s − 49-s − 24·59-s + 4·61-s − 64-s − 8·76-s + 32·79-s + 12·89-s + 12·101-s − 28·109-s + 12·116-s − 22·121-s + 8·124-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 + 1/4·16-s + 1.83·19-s − 2.22·29-s − 1.43·31-s − 1.87·41-s − 1/7·49-s − 3.12·59-s + 0.512·61-s − 1/8·64-s − 0.917·76-s + 3.60·79-s + 1.27·89-s + 1.19·101-s − 2.68·109-s + 1.11·116-s − 2·121-s + 0.718·124-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 + 0.937·164-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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.056477633$
$L(\frac12)$  $\approx$  $1.056477633$
$L(\frac{3}{2})$   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$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
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

−9.044553428327962660422903661411, −8.475599330455527378765883078711, −8.070456137943154175087844763333, −7.58573702554331648559733286775, −7.58218476486543759997009174505, −7.15401078043947886511583328200, −6.43265602712113626588566435157, −6.40995110952086286475092847198, −5.65829811136930735844716745151, −5.41163992768336156148376878162, −4.97876699452255437990135054285, −4.90978362149398567792015996993, −3.91047004660196640524981003685, −3.87879780623905568896066207502, −3.27108984758456766340236910427, −3.08532429289423874844650757683, −2.19972926975243127497532059098, −1.73758225492426653329847143699, −1.25579902994169803297794834921, −0.33278031377724664712408740075, 0.33278031377724664712408740075, 1.25579902994169803297794834921, 1.73758225492426653329847143699, 2.19972926975243127497532059098, 3.08532429289423874844650757683, 3.27108984758456766340236910427, 3.87879780623905568896066207502, 3.91047004660196640524981003685, 4.90978362149398567792015996993, 4.97876699452255437990135054285, 5.41163992768336156148376878162, 5.65829811136930735844716745151, 6.40995110952086286475092847198, 6.43265602712113626588566435157, 7.15401078043947886511583328200, 7.58218476486543759997009174505, 7.58573702554331648559733286775, 8.070456137943154175087844763333, 8.475599330455527378765883078711, 9.044553428327962660422903661411

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