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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.716287376$
$L(\frac12)$  $\approx$  $2.716287376$
$L(\frac{3}{2})$   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

−8.538725681242857869666908292632, −8.482748299997655803667908699491, −8.200121323913554069772071555711, −8.122393540218572032822276816006, −7.28002282381830107871109533855, −6.90582404766489243037836502391, −6.52300765544816658970992038916, −6.44244083100468576861231443861, −5.88210687213051791097291490817, −5.26336648189400356114605460107, −5.06083094473874519121372982960, −4.54944079256386358730642439459, −4.33758145465434388558654496862, −3.79068177135856846222756450670, −3.29071773038933164007453995990, −2.84891059370008941722530675145, −2.31612230040998051861296662072, −1.89248978017999300971331797570, −0.830246319784123610551317756921, −0.73863750808720447045536713662, 0.73863750808720447045536713662, 0.830246319784123610551317756921, 1.89248978017999300971331797570, 2.31612230040998051861296662072, 2.84891059370008941722530675145, 3.29071773038933164007453995990, 3.79068177135856846222756450670, 4.33758145465434388558654496862, 4.54944079256386358730642439459, 5.06083094473874519121372982960, 5.26336648189400356114605460107, 5.88210687213051791097291490817, 6.44244083100468576861231443861, 6.52300765544816658970992038916, 6.90582404766489243037836502391, 7.28002282381830107871109533855, 8.122393540218572032822276816006, 8.200121323913554069772071555711, 8.482748299997655803667908699491, 8.538725681242857869666908292632

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