Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{2} \cdot 29^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 4·5-s − 2·7-s + 9-s − 4·13-s + 5·16-s + 12·20-s + 2·25-s + 6·28-s − 2·29-s + 8·35-s − 3·36-s − 4·45-s + 3·49-s + 12·52-s + 12·53-s + 24·59-s − 2·63-s − 3·64-s + 16·65-s + 8·67-s − 20·80-s + 81-s − 24·83-s + 8·91-s − 6·100-s + 16·103-s + ⋯
L(s)  = 1  − 3/2·4-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.10·13-s + 5/4·16-s + 2.68·20-s + 2/5·25-s + 1.13·28-s − 0.371·29-s + 1.35·35-s − 1/2·36-s − 0.596·45-s + 3/7·49-s + 1.66·52-s + 1.64·53-s + 3.12·59-s − 0.251·63-s − 3/8·64-s + 1.98·65-s + 0.977·67-s − 2.23·80-s + 1/9·81-s − 2.63·83-s + 0.838·91-s − 3/5·100-s + 1.57·103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(370881\)    =    \(3^{2} \cdot 7^{2} \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{370881} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 370881,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;29\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 + T )^{2} \)
29$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.548328603682253116112277067930, −8.135249988088522958713590759000, −7.48292463845768060212531982747, −7.25047783802838427330028450541, −6.85540750527117050478666887987, −6.04718499105037929041596410377, −5.33127488666172597415900037702, −5.14083448762531515766594705626, −4.20755242463912897426431158801, −4.13559084050773741974089362056, −3.75007661297492683580937601463, −3.05279040294571468472224368289, −2.22318003493217483206616099006, −0.73756874722956271567483825317, 0, 0.73756874722956271567483825317, 2.22318003493217483206616099006, 3.05279040294571468472224368289, 3.75007661297492683580937601463, 4.13559084050773741974089362056, 4.20755242463912897426431158801, 5.14083448762531515766594705626, 5.33127488666172597415900037702, 6.04718499105037929041596410377, 6.85540750527117050478666887987, 7.25047783802838427330028450541, 7.48292463845768060212531982747, 8.135249988088522958713590759000, 8.548328603682253116112277067930

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