Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{6} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s − 2·11-s + 2·19-s − 2·23-s − 5·25-s − 2·31-s − 4·35-s + 6·37-s − 10·41-s + 4·43-s − 12·47-s + 3·49-s + 8·53-s + 4·55-s − 4·59-s + 8·67-s − 14·71-s − 12·73-s − 4·77-s − 4·79-s − 16·83-s + 6·89-s − 4·95-s − 8·97-s + 8·101-s − 2·103-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 0.603·11-s + 0.458·19-s − 0.417·23-s − 25-s − 0.359·31-s − 0.676·35-s + 0.986·37-s − 1.56·41-s + 0.609·43-s − 1.75·47-s + 3/7·49-s + 1.09·53-s + 0.539·55-s − 0.520·59-s + 0.977·67-s − 1.66·71-s − 1.40·73-s − 0.455·77-s − 0.450·79-s − 1.75·83-s + 0.635·89-s − 0.410·95-s − 0.812·97-s + 0.796·101-s − 0.197·103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36578304\)    =    \(2^{10} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 36578304,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 14 T + 189 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 154 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 169 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 178 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.72911258868049666759784880923, −7.70550271841845756538943195339, −7.25310505158434878971577178009, −7.08469061698709826964230035980, −6.32285309371567843876037682830, −6.25496498432210115765214637179, −5.61630816649468575215826929470, −5.44295654984685540592237392668, −4.97742731718137892331786805627, −4.62573260263317666763402331371, −4.14213487929080329600589400748, −4.02229390843710339940206698404, −3.28005564770321649212792263576, −3.24998250365293403974153238378, −2.41963106878474376545201733131, −2.23402238515108857774457074144, −1.43925972086987178386106677901, −1.20602011349772668883187851424, 0, 0, 1.20602011349772668883187851424, 1.43925972086987178386106677901, 2.23402238515108857774457074144, 2.41963106878474376545201733131, 3.24998250365293403974153238378, 3.28005564770321649212792263576, 4.02229390843710339940206698404, 4.14213487929080329600589400748, 4.62573260263317666763402331371, 4.97742731718137892331786805627, 5.44295654984685540592237392668, 5.61630816649468575215826929470, 6.25496498432210115765214637179, 6.32285309371567843876037682830, 7.08469061698709826964230035980, 7.25310505158434878971577178009, 7.70550271841845756538943195339, 7.72911258868049666759784880923

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