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  − 3·2-s + 5·4-s − 5·8-s + 9-s + 11-s − 4·13-s + 16-s − 3·18-s − 3·22-s − 2·23-s − 4·25-s + 12·26-s + 2·29-s + 7·32-s + 5·36-s − 18·41-s + 5·44-s + 6·46-s + 16·47-s + 9·49-s + 12·50-s − 20·52-s − 6·58-s − 15·64-s − 5·72-s + 6·79-s − 8·81-s + ⋯
L(s)  = 1  − 2.12·2-s + 5/2·4-s − 1.76·8-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1/4·16-s − 0.707·18-s − 0.639·22-s − 0.417·23-s − 4/5·25-s + 2.35·26-s + 0.371·29-s + 1.23·32-s + 5/6·36-s − 2.81·41-s + 0.753·44-s + 0.884·46-s + 2.33·47-s + 9/7·49-s + 1.69·50-s − 2.77·52-s − 0.787·58-s − 1.87·64-s − 0.589·72-s + 0.675·79-s − 8/9·81-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$ \( ( 1 + T )( 1 + p T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
11$C_2$ \( 1 - T + p T^{2} \)
19$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 47 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 145 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 171 T^{2} + p^{2} T^{4} \)
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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

−8.488165286437270730401594839756, −7.989949773300283141362357849796, −7.58160274808814998787496019415, −7.17302109493562102412958445666, −6.86111209817108375273552376011, −6.36046765793682467256567685736, −5.65074929653116029605763212523, −5.17115839866373868806154396628, −4.43120944678665957746113263556, −3.96898761787115612800662909995, −3.11390863068111940498613941325, −2.30292363832701496322963685469, −1.89462716154926679076231528824, −1.01380105590921296122083378756, 0, 1.01380105590921296122083378756, 1.89462716154926679076231528824, 2.30292363832701496322963685469, 3.11390863068111940498613941325, 3.96898761787115612800662909995, 4.43120944678665957746113263556, 5.17115839866373868806154396628, 5.65074929653116029605763212523, 6.36046765793682467256567685736, 6.86111209817108375273552376011, 7.17302109493562102412958445666, 7.58160274808814998787496019415, 7.989949773300283141362357849796, 8.488165286437270730401594839756

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