Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23^{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  + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 2·13-s − 14-s + 16-s − 6·17-s − 8·19-s + 20-s + 25-s + 2·26-s − 28-s − 6·29-s − 4·31-s + 32-s − 6·34-s − 35-s + 10·37-s − 8·38-s + 40-s + 6·41-s + 4·43-s + 49-s + 50-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 + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 1.83·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s + 1.64·37-s − 1.29·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(333270\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{333270} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 333270,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;23\}$, 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 \( 1 + T \)
23 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.87093369532272, −12.76803123826780, −11.89630553781175, −11.32663968598076, −11.14933821700255, −10.60944418748279, −10.30291310892982, −9.500123314786228, −9.255079264916240, −8.703135091322260, −8.282512483665025, −7.682098062069786, −7.033032673102099, −6.727757999212156, −6.166730790136605, −5.871715407156059, −5.422747340341407, −4.612718159910386, −4.198392276119098, −3.976917379035770, −3.206802545137822, −2.529641223639151, −2.187181210371254, −1.685314171478870, −0.7801536548888144, 0, 0.7801536548888144, 1.685314171478870, 2.187181210371254, 2.529641223639151, 3.206802545137822, 3.976917379035770, 4.198392276119098, 4.612718159910386, 5.422747340341407, 5.871715407156059, 6.166730790136605, 6.727757999212156, 7.033032673102099, 7.682098062069786, 8.282512483665025, 8.703135091322260, 9.255079264916240, 9.500123314786228, 10.30291310892982, 10.60944418748279, 11.14933821700255, 11.32663968598076, 11.89630553781175, 12.76803123826780, 12.87093369532272

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