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 + 20·29-s − 16·31-s + 4·41-s − 8·44-s − 49-s + 8·59-s − 12·61-s − 64-s + 24·71-s + 16·79-s + 28·89-s + 12·101-s + 36·109-s − 20·116-s + 26·121-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 + 2.41·11-s + 1/4·16-s + 3.71·29-s − 2.87·31-s + 0.624·41-s − 1.20·44-s − 1/7·49-s + 1.04·59-s − 1.53·61-s − 1/8·64-s + 2.84·71-s + 1.80·79-s + 2.96·89-s + 1.19·101-s + 3.44·109-s − 1.85·116-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 + 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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.426499116$
$L(\frac12)$  $\approx$  $3.426499116$
$L(\frac{3}{2})$   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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.762485760516814276506416481264, −8.749491786643779186228267231118, −8.128339392213194433757424584729, −7.88531183134421312445158632349, −7.16640032452598025229564295112, −7.10919006723036228969870917074, −6.50124166124116258540171234323, −6.22404062695061627230518680314, −6.10994024603543478578020940917, −5.34101421649244286684639540018, −4.92487744699090108147806507829, −4.65236891331094227497641237193, −4.18004133837923729013381197563, −3.60074446134783753497241225253, −3.58378121644805737314205261849, −2.96796623091637426906950349012, −2.12109098891343464276410383978, −1.84880430456966005618753217806, −0.927936750808387808129664658783, −0.78461377901731977779196383475, 0.78461377901731977779196383475, 0.927936750808387808129664658783, 1.84880430456966005618753217806, 2.12109098891343464276410383978, 2.96796623091637426906950349012, 3.58378121644805737314205261849, 3.60074446134783753497241225253, 4.18004133837923729013381197563, 4.65236891331094227497641237193, 4.92487744699090108147806507829, 5.34101421649244286684639540018, 6.10994024603543478578020940917, 6.22404062695061627230518680314, 6.50124166124116258540171234323, 7.10919006723036228969870917074, 7.16640032452598025229564295112, 7.88531183134421312445158632349, 8.128339392213194433757424584729, 8.749491786643779186228267231118, 8.762485760516814276506416481264

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