Properties

Degree 4
Conductor $ 2 \cdot 5 \cdot 11^{2} \cdot 19^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 3·5-s − 2·8-s − 2·9-s + 4·11-s − 4·13-s + 16-s − 6·19-s + 3·20-s − 2·23-s + 8·25-s + 2·29-s + 4·32-s + 2·36-s + 6·40-s − 4·44-s + 6·45-s + 16·47-s + 12·49-s + 4·52-s − 12·55-s + 3·64-s + 12·65-s + 4·72-s + 6·76-s + 12·79-s − 3·80-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.34·5-s − 0.707·8-s − 2/3·9-s + 1.20·11-s − 1.10·13-s + 1/4·16-s − 1.37·19-s + 0.670·20-s − 0.417·23-s + 8/5·25-s + 0.371·29-s + 0.707·32-s + 1/3·36-s + 0.948·40-s − 0.603·44-s + 0.894·45-s + 2.33·47-s + 12/7·49-s + 0.554·52-s − 1.61·55-s + 3/8·64-s + 1.48·65-s + 0.471·72-s + 0.688·76-s + 1.35·79-s − 0.335·80-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 436810 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 436810 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(436810\)    =    \(2 \cdot 5 \cdot 11^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{436810} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 436810,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;11,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
19$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
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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

−8.494983984321189824551705515643, −7.941775096355573751347969564559, −7.55898125824351236541292342852, −6.90707784074107163787605156377, −6.68424449399363793636758375745, −6.00295031540585248865373829650, −5.59445767169428069519305601113, −4.90864643316707902464148549083, −4.34931844773047449558933574180, −4.01210086911151176460804617939, −3.57399221264695649491758463932, −2.74851954279350005587494526570, −2.31510196794726164373319215236, −0.921549437007102636044542491351, 0, 0.921549437007102636044542491351, 2.31510196794726164373319215236, 2.74851954279350005587494526570, 3.57399221264695649491758463932, 4.01210086911151176460804617939, 4.34931844773047449558933574180, 4.90864643316707902464148549083, 5.59445767169428069519305601113, 6.00295031540585248865373829650, 6.68424449399363793636758375745, 6.90707784074107163787605156377, 7.55898125824351236541292342852, 7.941775096355573751347969564559, 8.494983984321189824551705515643

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