Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{4} \cdot 41^{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  + 2·3-s − 4·4-s + 3·9-s + 8·11-s − 8·12-s − 4·13-s + 12·16-s − 10·17-s − 8·19-s + 4·23-s + 2·25-s + 4·27-s − 8·31-s + 16·33-s − 12·36-s − 2·37-s − 8·39-s + 2·41-s + 6·43-s − 32·44-s + 10·47-s + 24·48-s − 20·51-s + 16·52-s + 8·53-s − 16·57-s − 32·64-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s + 9-s + 2.41·11-s − 2.30·12-s − 1.10·13-s + 3·16-s − 2.42·17-s − 1.83·19-s + 0.834·23-s + 2/5·25-s + 0.769·27-s − 1.43·31-s + 2.78·33-s − 2·36-s − 0.328·37-s − 1.28·39-s + 0.312·41-s + 0.914·43-s − 4.82·44-s + 1.45·47-s + 3.46·48-s − 2.80·51-s + 2.21·52-s + 1.09·53-s − 2.11·57-s − 4·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36324729\)    =    \(3^{2} \cdot 7^{4} \cdot 41^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6027} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 36324729,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.323738458$
$L(\frac12)$  $\approx$  $2.323738458$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;41\}$, 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 \( 1 \)
41$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 119 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 179 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 135 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−8.353678608770662847299865380460, −8.332084222414555977303129528652, −7.55720524177333256665773263579, −7.09519376516687918696654176803, −6.89885473750085340446559748583, −6.76046695167375716021199324542, −6.07985993550982072504098221293, −5.75050050996984274339697217848, −5.28906681727335924511498607093, −4.70148630756669300876138364389, −4.37617718652479557286822383532, −4.34428163346596125717697206755, −3.82311169850449025043110520212, −3.80748329693678630939012360002, −3.08415766596524700145753616135, −2.57498038315691314110759528709, −1.96474480712990866720892942998, −1.80093794891218194311150207312, −0.887454135928322001658700104375, −0.46538309134254912408897046436, 0.46538309134254912408897046436, 0.887454135928322001658700104375, 1.80093794891218194311150207312, 1.96474480712990866720892942998, 2.57498038315691314110759528709, 3.08415766596524700145753616135, 3.80748329693678630939012360002, 3.82311169850449025043110520212, 4.34428163346596125717697206755, 4.37617718652479557286822383532, 4.70148630756669300876138364389, 5.28906681727335924511498607093, 5.75050050996984274339697217848, 6.07985993550982072504098221293, 6.76046695167375716021199324542, 6.89885473750085340446559748583, 7.09519376516687918696654176803, 7.55720524177333256665773263579, 8.332084222414555977303129528652, 8.353678608770662847299865380460

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