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 − 4·19-s − 12·29-s + 16·31-s + 12·41-s − 49-s − 24·59-s + 16·61-s − 64-s + 12·71-s + 4·76-s + 32·79-s − 12·89-s + 12·101-s + 20·109-s + 12·116-s − 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/4·16-s − 0.917·19-s − 2.22·29-s + 2.87·31-s + 1.87·41-s − 1/7·49-s − 3.12·59-s + 2.04·61-s − 1/8·64-s + 1.42·71-s + 0.458·76-s + 3.60·79-s − 1.27·89-s + 1.19·101-s + 1.91·109-s + 1.11·116-s − 2·121-s − 1.43·124-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 + ⋯

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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.778206092$
$L(\frac12)$  $\approx$  $1.778206092$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.913588742562795045312389790636, −8.489250078825845077535406397302, −7.969147393638267170447910365320, −7.915321324046354594827067599451, −7.53524729916976273244206017927, −6.96553250866301347378844342410, −6.53678653285585313234974678592, −6.16171876502182176338580026076, −6.01343750212412150270780751421, −5.33040325595765643015636409375, −5.06586964490221257497733588172, −4.44042058937328080571874913564, −4.39717522695574308172824452433, −3.59135396476997925927508395011, −3.58830600477777672384790510525, −2.68763404797971428976602987156, −2.40296417458936995675272446746, −1.82744132002226770271992457725, −1.07208474722688295718326417191, −0.47436319440764993860101528223, 0.47436319440764993860101528223, 1.07208474722688295718326417191, 1.82744132002226770271992457725, 2.40296417458936995675272446746, 2.68763404797971428976602987156, 3.58830600477777672384790510525, 3.59135396476997925927508395011, 4.39717522695574308172824452433, 4.44042058937328080571874913564, 5.06586964490221257497733588172, 5.33040325595765643015636409375, 6.01343750212412150270780751421, 6.16171876502182176338580026076, 6.53678653285585313234974678592, 6.96553250866301347378844342410, 7.53524729916976273244206017927, 7.915321324046354594827067599451, 7.969147393638267170447910365320, 8.489250078825845077535406397302, 8.913588742562795045312389790636

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