Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{6} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·7-s + 2·11-s − 9·13-s + 2·19-s + 5·23-s − 5·25-s − 15·29-s + 31-s − 2·35-s + 3·37-s + 19·41-s − 2·43-s − 3·47-s + 3·49-s + 53-s − 2·55-s − 8·59-s + 9·65-s − 19·67-s − 71-s − 6·73-s + 4·77-s − 13·79-s + 83-s + 15·89-s − 18·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s + 0.603·11-s − 2.49·13-s + 0.458·19-s + 1.04·23-s − 25-s − 2.78·29-s + 0.179·31-s − 0.338·35-s + 0.493·37-s + 2.96·41-s − 0.304·43-s − 0.437·47-s + 3/7·49-s + 0.137·53-s − 0.269·55-s − 1.04·59-s + 1.11·65-s − 2.32·67-s − 0.118·71-s − 0.702·73-s + 0.455·77-s − 1.46·79-s + 0.109·83-s + 1.58·89-s − 1.88·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36578304\)    =    \(2^{10} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 36578304,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 9 T + 42 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 15 T + 110 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 19 T + 168 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - T + 68 T^{2} - p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 19 T + 186 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 138 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 13 T + 196 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - T + 162 T^{2} - p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 15 T + 196 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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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

−7.69086192831378548868958567257, −7.46041479001014593241478061909, −7.26444506304166993370996765374, −7.25803027792953015123843355978, −6.43283918930447401889335975553, −6.14943386623009137089814164984, −5.56815877437785711257551015919, −5.42192111246663128945299226973, −5.08943338219129930667223497138, −4.48185979240217685196597872859, −4.22009130588586958134212661837, −4.13330052865976744139058625158, −3.37997415252456926925137901879, −2.98686043228372546323706324984, −2.38487902533441043497525122208, −2.32016830210195000109132112181, −1.43942109295074372590213029140, −1.24966116164055530999661264810, 0, 0, 1.24966116164055530999661264810, 1.43942109295074372590213029140, 2.32016830210195000109132112181, 2.38487902533441043497525122208, 2.98686043228372546323706324984, 3.37997415252456926925137901879, 4.13330052865976744139058625158, 4.22009130588586958134212661837, 4.48185979240217685196597872859, 5.08943338219129930667223497138, 5.42192111246663128945299226973, 5.56815877437785711257551015919, 6.14943386623009137089814164984, 6.43283918930447401889335975553, 7.25803027792953015123843355978, 7.26444506304166993370996765374, 7.46041479001014593241478061909, 7.69086192831378548868958567257

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