Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{8} \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  − 7-s + 6·11-s − 2·13-s − 8·19-s + 6·23-s + 5·25-s − 6·29-s − 8·31-s + 4·37-s − 12·41-s + 4·43-s − 12·47-s − 12·53-s + 10·61-s − 8·67-s + 12·71-s − 20·73-s − 6·77-s + 4·79-s + 12·83-s + 24·89-s + 2·91-s + 10·97-s + 12·101-s − 8·103-s − 12·107-s + 28·109-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.80·11-s − 0.554·13-s − 1.83·19-s + 1.25·23-s + 25-s − 1.11·29-s − 1.43·31-s + 0.657·37-s − 1.87·41-s + 0.609·43-s − 1.75·47-s − 1.64·53-s + 1.28·61-s − 0.977·67-s + 1.42·71-s − 2.34·73-s − 0.683·77-s + 0.450·79-s + 1.31·83-s + 2.54·89-s + 0.209·91-s + 1.01·97-s + 1.19·101-s − 0.788·103-s − 1.16·107-s + 2.68·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5143824\)    =    \(2^{4} \cdot 3^{8} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2268} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 5143824,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.599163012\)
\(L(\frac12)\)  \(\approx\)  \(1.599163012\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−9.277514125400977374924380205898, −8.916673708942122618435374206025, −8.581633392470210362959985492589, −8.139114171419124658855805662152, −7.55487425682306110698043040046, −7.26718065801961995198100705756, −6.80280541089586687415480976047, −6.45421799154760728205789787926, −6.31189553496052650422038856896, −5.80930879213329643422588999269, −5.04670819917669227594888150587, −4.87651503975233622159056832727, −4.46884447186603265294106039120, −3.71241073276804783745447111211, −3.61902159725980669399051113580, −3.11662560351249626213377973979, −2.36346957374264415656614906676, −1.83537165811994897304706491537, −1.39522684320367510089473238635, −0.43932082541865337390556094970, 0.43932082541865337390556094970, 1.39522684320367510089473238635, 1.83537165811994897304706491537, 2.36346957374264415656614906676, 3.11662560351249626213377973979, 3.61902159725980669399051113580, 3.71241073276804783745447111211, 4.46884447186603265294106039120, 4.87651503975233622159056832727, 5.04670819917669227594888150587, 5.80930879213329643422588999269, 6.31189553496052650422038856896, 6.45421799154760728205789787926, 6.80280541089586687415480976047, 7.26718065801961995198100705756, 7.55487425682306110698043040046, 8.139114171419124658855805662152, 8.581633392470210362959985492589, 8.916673708942122618435374206025, 9.277514125400977374924380205898

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