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 + 12·29-s − 16·31-s − 4·41-s + 8·44-s − 49-s + 8·59-s − 4·61-s − 64-s − 16·71-s − 8·76-s + 4·89-s − 12·101-s + 4·109-s − 12·116-s + 26·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s − 2.41·11-s + 1/4·16-s + 1.83·19-s + 2.22·29-s − 2.87·31-s − 0.624·41-s + 1.20·44-s − 1/7·49-s + 1.04·59-s − 0.512·61-s − 1/8·64-s − 1.89·71-s − 0.917·76-s + 0.423·89-s − 1.19·101-s + 0.383·109-s − 1.11·116-s + 2.36·121-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 + ⋯

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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.7502964889$
$L(\frac12)$  $\approx$  $0.7502964889$
$L(\frac{3}{2})$   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$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−8.906932641626885430617665078138, −8.416001800746905333477650856511, −8.186374640050119127973168262770, −7.56772470458726805704817209972, −7.45755297410796835159792071799, −7.34774345659382178971157875641, −6.53789868662020627382767062649, −6.26936768074248212635750847128, −5.57725204217112317878155124369, −5.35136070724991637364620316498, −5.09125125091688346379439405377, −4.89686538493475803700876431381, −4.13987840379394228599661793466, −3.78523889567437294335980661957, −3.14175913505525784010171094206, −2.88546078775332509877633844895, −2.49762955761122481548485427500, −1.75103076499376716873229677080, −1.14614896765759484954306572506, −0.29039028538893551807039851917, 0.29039028538893551807039851917, 1.14614896765759484954306572506, 1.75103076499376716873229677080, 2.49762955761122481548485427500, 2.88546078775332509877633844895, 3.14175913505525784010171094206, 3.78523889567437294335980661957, 4.13987840379394228599661793466, 4.89686538493475803700876431381, 5.09125125091688346379439405377, 5.35136070724991637364620316498, 5.57725204217112317878155124369, 6.26936768074248212635750847128, 6.53789868662020627382767062649, 7.34774345659382178971157875641, 7.45755297410796835159792071799, 7.56772470458726805704817209972, 8.186374640050119127973168262770, 8.416001800746905333477650856511, 8.906932641626885430617665078138

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