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 − 12·19-s − 12·29-s − 8·31-s − 20·41-s − 8·44-s − 49-s − 8·59-s − 16·61-s − 64-s + 4·71-s + 12·76-s − 32·79-s + 4·89-s + 12·101-s + 36·109-s + 12·116-s + 26·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s + 2.41·11-s + 1/4·16-s − 2.75·19-s − 2.22·29-s − 1.43·31-s − 3.12·41-s − 1.20·44-s − 1/7·49-s − 1.04·59-s − 2.04·61-s − 1/8·64-s + 0.474·71-s + 1.37·76-s − 3.60·79-s + 0.423·89-s + 1.19·101-s + 3.44·109-s + 1.11·116-s + 2.36·121-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 + ⋯

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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3579095154$
$L(\frac12)$  $\approx$  $0.3579095154$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−8.989944523851944745677400490040, −8.551688120957419184863446133736, −8.452638401522782125996941855119, −7.57417570802799638399813567576, −7.43204772394178331423441551924, −6.99427010476788893980956406078, −6.51695129391570008242047856145, −6.21935800601662952174837603363, −6.03471319516934027341951167081, −5.49338183265968951560534288483, −4.84336855542512028900222161397, −4.64343749736367863339587409196, −4.04741547347709638169950482970, −3.80749737426478249365233225326, −3.55794752597154190458651212811, −2.95592546566307436550590342399, −2.02369591564810194406885870016, −1.69788553547637432017326320021, −1.47430329652078000632173265111, −0.18014242094875388600744436718, 0.18014242094875388600744436718, 1.47430329652078000632173265111, 1.69788553547637432017326320021, 2.02369591564810194406885870016, 2.95592546566307436550590342399, 3.55794752597154190458651212811, 3.80749737426478249365233225326, 4.04741547347709638169950482970, 4.64343749736367863339587409196, 4.84336855542512028900222161397, 5.49338183265968951560534288483, 6.03471319516934027341951167081, 6.21935800601662952174837603363, 6.51695129391570008242047856145, 6.99427010476788893980956406078, 7.43204772394178331423441551924, 7.57417570802799638399813567576, 8.452638401522782125996941855119, 8.551688120957419184863446133736, 8.989944523851944745677400490040

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