Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 19^{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 − 4·7-s − 8-s − 10-s − 2·13-s + 4·14-s + 16-s − 6·17-s + 20-s + 25-s + 2·26-s − 4·28-s − 6·29-s − 8·31-s − 32-s + 6·34-s − 4·35-s − 2·37-s − 40-s − 6·41-s − 4·43-s + 9·49-s − 50-s − 2·52-s − 6·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.755·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.676·35-s − 0.328·37-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.141·50-s − 0.277·52-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(32490\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{32490} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((2,\ 32490,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
19 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
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

−15.68871891855849, −15.13057775903490, −14.68518867848415, −13.87992790883157, −13.35802426434237, −12.86660415454204, −12.56665288933569, −11.77537623885046, −11.23026605287229, −10.59800939405654, −10.19182791619861, −9.520735329122987, −9.196600711554548, −8.839330674022020, −7.958790934135412, −7.327073450451754, −6.670242106999230, −6.524188486789435, −5.684073447136196, −5.146590038241415, −4.195465799735141, −3.500051894332702, −2.864515048081545, −2.165258349072296, −1.506294887114446, 0, 0, 1.506294887114446, 2.165258349072296, 2.864515048081545, 3.500051894332702, 4.195465799735141, 5.146590038241415, 5.684073447136196, 6.524188486789435, 6.670242106999230, 7.327073450451754, 7.958790934135412, 8.839330674022020, 9.196600711554548, 9.520735329122987, 10.19182791619861, 10.59800939405654, 11.23026605287229, 11.77537623885046, 12.56665288933569, 12.86660415454204, 13.35802426434237, 13.87992790883157, 14.68518867848415, 15.13057775903490, 15.68871891855849

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