Properties

Degree $4$
Conductor $2286144$
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  − 2·5-s + 7-s − 4·13-s + 6·17-s + 19-s − 2·23-s + 5·25-s + 12·29-s − 31-s − 2·35-s − 2·37-s + 4·41-s + 18·43-s + 2·47-s − 6·49-s + 6·53-s + 8·59-s − 11·61-s + 8·65-s + 12·67-s + 8·71-s − 5·73-s + 4·79-s + 8·83-s − 12·85-s − 18·89-s − 4·91-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 1.10·13-s + 1.45·17-s + 0.229·19-s − 0.417·23-s + 25-s + 2.22·29-s − 0.179·31-s − 0.338·35-s − 0.328·37-s + 0.624·41-s + 2.74·43-s + 0.291·47-s − 6/7·49-s + 0.824·53-s + 1.04·59-s − 1.40·61-s + 0.992·65-s + 1.46·67-s + 0.949·71-s − 0.585·73-s + 0.450·79-s + 0.878·83-s − 1.30·85-s − 1.90·89-s − 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1512} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.092462703\)
\(L(\frac12)\) \(\approx\) \(2.092462703\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.798295131946900679255190336462, −9.328452850962182473162756689369, −8.694665748065900063379766906093, −8.516394905834565590537238602637, −7.930409222946531544087263450389, −7.76348952180550052799532236895, −7.31684744172923089916144354756, −7.02293442072793526739936197948, −6.48800632249584503681377331316, −5.94503026095327874346981430824, −5.50816857353372588428262268864, −5.09134814278855467995815352418, −4.53062754939942534503311676295, −4.34679744044240182802171336437, −3.66675731695643390722182114447, −3.20222581647745350566525667192, −2.64211136845691535474448416617, −2.21503541101929305742060862499, −1.14381699843739766241159282098, −0.67869172730049211863501936009, 0.67869172730049211863501936009, 1.14381699843739766241159282098, 2.21503541101929305742060862499, 2.64211136845691535474448416617, 3.20222581647745350566525667192, 3.66675731695643390722182114447, 4.34679744044240182802171336437, 4.53062754939942534503311676295, 5.09134814278855467995815352418, 5.50816857353372588428262268864, 5.94503026095327874346981430824, 6.48800632249584503681377331316, 7.02293442072793526739936197948, 7.31684744172923089916144354756, 7.76348952180550052799532236895, 7.930409222946531544087263450389, 8.516394905834565590537238602637, 8.694665748065900063379766906093, 9.328452850962182473162756689369, 9.798295131946900679255190336462

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