Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{2} \cdot 11^{4} $
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  − 2·3-s + 4-s + 5-s − 2·7-s + 3·9-s − 2·12-s − 2·13-s − 2·15-s − 3·16-s − 3·17-s + 19-s + 20-s + 4·21-s + 11·23-s − 8·25-s − 4·27-s − 2·28-s + 12·29-s + 5·31-s − 2·35-s + 3·36-s + 7·37-s + 4·39-s − 17·41-s − 6·43-s + 3·45-s + 6·48-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 9-s − 0.577·12-s − 0.554·13-s − 0.516·15-s − 3/4·16-s − 0.727·17-s + 0.229·19-s + 0.223·20-s + 0.872·21-s + 2.29·23-s − 8/5·25-s − 0.769·27-s − 0.377·28-s + 2.22·29-s + 0.898·31-s − 0.338·35-s + 1/2·36-s + 1.15·37-s + 0.640·39-s − 2.65·41-s − 0.914·43-s + 0.447·45-s + 0.866·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6456681\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2541} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 6456681,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.229234278$
$L(\frac12)$  $\approx$  $1.229234278$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 11 T + 75 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 7 T + 85 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 17 T + 153 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 9 T + 197 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.023743568221924504216409428378, −8.848433022841399156531359095853, −8.355754819927098073687663906451, −7.914415105534707878010448230928, −7.34809756143244920901064055098, −6.90895612382047214695359970370, −6.72716092899351277528219788226, −6.51921128365278261715880817369, −6.18082924958431772662663023560, −5.50879013082944547998349927684, −5.18154083423738583247014252208, −4.99845124870622948749510710670, −4.23853109921884502856414249422, −4.18039010281958255531008494955, −3.28835259961409353657930448881, −2.76179934789303761200821768323, −2.53682875350577112771699235860, −1.78910380924120353574255154314, −1.15114229669994717867736904528, −0.42898484245146638831882434050, 0.42898484245146638831882434050, 1.15114229669994717867736904528, 1.78910380924120353574255154314, 2.53682875350577112771699235860, 2.76179934789303761200821768323, 3.28835259961409353657930448881, 4.18039010281958255531008494955, 4.23853109921884502856414249422, 4.99845124870622948749510710670, 5.18154083423738583247014252208, 5.50879013082944547998349927684, 6.18082924958431772662663023560, 6.51921128365278261715880817369, 6.72716092899351277528219788226, 6.90895612382047214695359970370, 7.34809756143244920901064055098, 7.914415105534707878010448230928, 8.355754819927098073687663906451, 8.848433022841399156531359095853, 9.023743568221924504216409428378

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