Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s − 2·13-s + 16-s + 6·17-s + 4·19-s − 20-s + 25-s − 2·26-s + 6·29-s − 8·31-s + 32-s + 6·34-s + 2·37-s + 4·38-s − 40-s − 6·41-s − 4·43-s + 50-s − 2·52-s + 6·53-s + 6·58-s + 10·61-s − 8·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 0.141·50-s − 0.277·52-s + 0.824·53-s + 0.787·58-s + 1.28·61-s − 1.01·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4410} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4410,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.925820309\)
\(L(\frac12)\)  \(\approx\)  \(2.925820309\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.03872433773004, −17.19004551940908, −16.49179303853663, −16.19574749973847, −15.41908166910923, −14.72843684638199, −14.40594824216390, −13.65438857725626, −13.01886305707585, −12.20745253833249, −11.95018776739917, −11.27452323153489, −10.37055942266627, −9.923251475716514, −9.044892677337592, −8.145462184294923, −7.518289536043852, −6.979136675139992, −6.037955731219678, −5.265701536554312, −4.757015561352326, −3.633594455313244, −3.218266564774695, −2.101333453055154, −0.8798172800695237, 0.8798172800695237, 2.101333453055154, 3.218266564774695, 3.633594455313244, 4.757015561352326, 5.265701536554312, 6.037955731219678, 6.979136675139992, 7.518289536043852, 8.145462184294923, 9.044892677337592, 9.923251475716514, 10.37055942266627, 11.27452323153489, 11.95018776739917, 12.20745253833249, 13.01886305707585, 13.65438857725626, 14.40594824216390, 14.72843684638199, 15.41908166910923, 16.19574749973847, 16.49179303853663, 17.19004551940908, 18.03872433773004

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