Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{6} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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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  =  0
Selberg data  =  $(4,\ 36578304,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $4.967758692$
$L(\frac12)$  $\approx$  $4.967758692$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
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

−8.059075352856134069543476525485, −7.955874635261739196483431363103, −7.49064815619819419076750305311, −7.36709344465195516789432586029, −6.71326673831289826208637828118, −6.53303561531071919140966318193, −5.92998171761605678684043751646, −5.88891211746463901855552092309, −5.33341191471367416503988222433, −5.23807558160348150480726846779, −4.52777123656924177146015015797, −4.30570054407303929902881309333, −3.87373401442183221174782868856, −3.55583655133397246698240256288, −2.79033540962842159319533932106, −2.56144907753032378553189460721, −2.04420500252458323822025892120, −1.65771943504029432327298600489, −1.07195962188983877032879864374, −0.62793247498384538704403764775, 0.62793247498384538704403764775, 1.07195962188983877032879864374, 1.65771943504029432327298600489, 2.04420500252458323822025892120, 2.56144907753032378553189460721, 2.79033540962842159319533932106, 3.55583655133397246698240256288, 3.87373401442183221174782868856, 4.30570054407303929902881309333, 4.52777123656924177146015015797, 5.23807558160348150480726846779, 5.33341191471367416503988222433, 5.88891211746463901855552092309, 5.92998171761605678684043751646, 6.53303561531071919140966318193, 6.71326673831289826208637828118, 7.36709344465195516789432586029, 7.49064815619819419076750305311, 7.955874635261739196483431363103, 8.059075352856134069543476525485

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