Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 4·11-s − 2·13-s − 14-s + 16-s + 4·19-s + 20-s + 4·22-s − 8·23-s + 25-s + 2·26-s + 28-s − 2·29-s − 32-s + 35-s − 6·37-s − 4·38-s − 40-s − 6·41-s − 4·43-s − 4·44-s + 8·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.917·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.176·32-s + 0.169·35-s − 0.986·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.603·44-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(182070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{182070} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((2,\ 182070,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;17\}$, 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 \( 1 - T \)
17 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.69546150540905, −13.30927870516156, −12.45018584657742, −12.18667864264021, −11.76162640847625, −11.21507282538194, −10.56063222073396, −10.27068488334725, −9.949913738458962, −9.427369194419357, −8.758191597939948, −8.491518716792713, −7.746729366664906, −7.527013007663030, −7.107512072858169, −6.313784086912772, −5.807576028210405, −5.436476380763992, −4.839179925065431, −4.323344208079482, −3.456331145783922, −2.924815236998421, −2.397468100729080, −1.703591885312310, −1.353017717621134, 0, 0, 1.353017717621134, 1.703591885312310, 2.397468100729080, 2.924815236998421, 3.456331145783922, 4.323344208079482, 4.839179925065431, 5.436476380763992, 5.807576028210405, 6.313784086912772, 7.107512072858169, 7.527013007663030, 7.746729366664906, 8.491518716792713, 8.758191597939948, 9.427369194419357, 9.949913738458962, 10.27068488334725, 10.56063222073396, 11.21507282538194, 11.76162640847625, 12.18667864264021, 12.45018584657742, 13.30927870516156, 13.69546150540905

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