Properties

Degree 4
Conductor $ 2 \cdot 3^{4} \cdot 5 $
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 + 3-s − 4-s − 5-s − 6-s − 5·7-s + 8-s + 9-s + 10-s − 12-s + 7·13-s + 5·14-s − 15-s + 3·16-s + 6·17-s − 18-s − 11·19-s + 20-s − 5·21-s + 24-s − 4·25-s − 7·26-s + 27-s + 5·28-s − 6·29-s + 30-s + 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 1.94·13-s + 1.33·14-s − 0.258·15-s + 3/4·16-s + 1.45·17-s − 0.235·18-s − 2.52·19-s + 0.223·20-s − 1.09·21-s + 0.204·24-s − 4/5·25-s − 1.37·26-s + 0.192·27-s + 0.944·28-s − 1.11·29-s + 0.182·30-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{810} (1, \cdot )$
Sato-Tate  :  $N(G_{1,3})$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 810,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3289823157$
$L(\frac12)$  $\approx$  $0.3289823157$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−19.5076338893, −19.1626037497, −18.6983747433, −18.5592411054, −17.5495821026, −16.9011141517, −16.3000171353, −16.0225385141, −14.969101086, −14.8921107674, −13.5821136983, −13.3351916393, −12.7156394907, −12.1436052702, −10.908728293, −10.3916050944, −9.42919920821, −9.30558712287, −8.21765036746, −7.94123132226, −6.42217617653, −6.04893540001, −4.0430440138, −3.36585804146, 3.36585804146, 4.0430440138, 6.04893540001, 6.42217617653, 7.94123132226, 8.21765036746, 9.30558712287, 9.42919920821, 10.3916050944, 10.908728293, 12.1436052702, 12.7156394907, 13.3351916393, 13.5821136983, 14.8921107674, 14.969101086, 16.0225385141, 16.3000171353, 16.9011141517, 17.5495821026, 18.5592411054, 18.6983747433, 19.1626037497, 19.5076338893

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