Properties

Degree 4
Conductor $ 3^{6} \cdot 5^{2} \cdot 7^{2} $
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-s + 2·5-s + 2·7-s + 8-s + 2·10-s + 6·11-s − 13-s + 2·14-s − 16-s − 17-s − 3·19-s + 6·22-s + 5·23-s + 3·25-s − 26-s + 7·29-s − 12·31-s − 6·32-s − 34-s + 4·35-s − 3·38-s + 2·40-s − 3·41-s + 6·43-s + 5·46-s + 8·47-s + 3·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.894·5-s + 0.755·7-s + 0.353·8-s + 0.632·10-s + 1.80·11-s − 0.277·13-s + 0.534·14-s − 1/4·16-s − 0.242·17-s − 0.688·19-s + 1.27·22-s + 1.04·23-s + 3/5·25-s − 0.196·26-s + 1.29·29-s − 2.15·31-s − 1.06·32-s − 0.171·34-s + 0.676·35-s − 0.486·38-s + 0.316·40-s − 0.468·41-s + 0.914·43-s + 0.737·46-s + 1.16·47-s + 3/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(893025\)    =    \(3^{6} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{945} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 893025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $4.54836$
$L(\frac12)$  $\approx$  $4.54836$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 31 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 5 T + 49 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 55 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T - 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 15 T + 169 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 23 T + 261 T^{2} + 23 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 10 T + 139 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 113 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−10.32311307127426569087854726205, −9.861698593090997841796095439340, −9.168744348213449786245213795433, −9.157573622441697866832827370646, −8.671805266307598352259911950727, −8.379419098498571764255089098175, −7.54840750603366428197244184178, −7.00208684727925169722721474597, −6.99399019176353753734224880731, −6.41516095465803900924521091338, −5.69420788193719833106852752689, −5.61599385168131930874559514651, −4.80938751146346007031929738094, −4.64584271661518399921731908072, −3.97193850209979659721996294033, −3.76206591111235032074363160548, −2.83922855426164383936617844541, −2.16773271275343725080656488029, −1.69463971121385263586712241401, −0.978012379483523488823020879447, 0.978012379483523488823020879447, 1.69463971121385263586712241401, 2.16773271275343725080656488029, 2.83922855426164383936617844541, 3.76206591111235032074363160548, 3.97193850209979659721996294033, 4.64584271661518399921731908072, 4.80938751146346007031929738094, 5.61599385168131930874559514651, 5.69420788193719833106852752689, 6.41516095465803900924521091338, 6.99399019176353753734224880731, 7.00208684727925169722721474597, 7.54840750603366428197244184178, 8.379419098498571764255089098175, 8.671805266307598352259911950727, 9.157573622441697866832827370646, 9.168744348213449786245213795433, 9.861698593090997841796095439340, 10.32311307127426569087854726205

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