Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{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  + 6·5-s + 7-s + 6·11-s + 12·17-s − 7·19-s + 19·25-s − 5·31-s + 6·35-s − 37-s − 6·47-s − 6·49-s + 36·55-s + 3·67-s + 15·73-s + 6·77-s − 27·79-s + 12·83-s + 72·85-s − 12·89-s − 42·95-s + 6·101-s − 5·103-s + 12·107-s − 5·109-s − 12·113-s + 12·119-s + 13·121-s + ⋯
L(s)  = 1  + 2.68·5-s + 0.377·7-s + 1.80·11-s + 2.91·17-s − 1.60·19-s + 19/5·25-s − 0.898·31-s + 1.01·35-s − 0.164·37-s − 0.875·47-s − 6/7·49-s + 4.85·55-s + 0.366·67-s + 1.75·73-s + 0.683·77-s − 3.03·79-s + 1.31·83-s + 7.80·85-s − 1.27·89-s − 4.30·95-s + 0.597·101-s − 0.492·103-s + 1.16·107-s − 0.478·109-s − 1.12·113-s + 1.10·119-s + 1.18·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.880719922\)
\(L(\frac12)\)  \(\approx\)  \(4.880719922\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 27 T + 322 T^{2} + 27 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−9.926643816528771425453593935499, −9.800331275349243628524798112220, −9.460921995055035657665285988072, −9.155022842141244979940517030927, −8.476465864121345578363289494351, −8.383216908065621612979749664439, −7.62190648936649790895299040758, −7.11363521738947374123338657465, −6.60451195973772045311178607638, −6.22688278065087222181262375563, −5.95874539460402550803226175504, −5.53337107360766385145487256840, −5.15166699712721117624005442623, −4.61482375082979772642949919702, −3.81087117394956968277353124733, −3.45105511310818678399443460005, −2.67290374323381343025240812578, −2.02841478795837421035871178007, −1.47271968346938712348794827883, −1.24848984411413032704653694327, 1.24848984411413032704653694327, 1.47271968346938712348794827883, 2.02841478795837421035871178007, 2.67290374323381343025240812578, 3.45105511310818678399443460005, 3.81087117394956968277353124733, 4.61482375082979772642949919702, 5.15166699712721117624005442623, 5.53337107360766385145487256840, 5.95874539460402550803226175504, 6.22688278065087222181262375563, 6.60451195973772045311178607638, 7.11363521738947374123338657465, 7.62190648936649790895299040758, 8.383216908065621612979749664439, 8.476465864121345578363289494351, 9.155022842141244979940517030927, 9.460921995055035657665285988072, 9.800331275349243628524798112220, 9.926643816528771425453593935499

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