Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 11 \cdot 17^{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 − 2·7-s + 8-s − 11-s − 4·13-s − 2·14-s + 16-s − 4·19-s − 22-s + 6·23-s − 5·25-s − 4·26-s − 2·28-s + 6·29-s − 8·31-s + 32-s + 10·37-s − 4·38-s + 6·41-s + 8·43-s − 44-s + 6·46-s + 6·47-s − 3·49-s − 5·50-s − 4·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.301·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.917·19-s − 0.213·22-s + 1.25·23-s − 25-s − 0.784·26-s − 0.377·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 1.64·37-s − 0.648·38-s + 0.937·41-s + 1.21·43-s − 0.150·44-s + 0.884·46-s + 0.875·47-s − 3/7·49-s − 0.707·50-s − 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(57222\)    =    \(2 \cdot 3^{2} \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{57222} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 57222,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
11 \( 1 + T \)
17 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.63917718491213, −14.14118717436648, −13.53509834236016, −13.01185763411863, −12.64584504154000, −12.31510876453235, −11.68012595404732, −10.93322720466951, −10.76367957690681, −10.01964282591604, −9.431420879070070, −9.161810285238644, −8.283806951536278, −7.671802963374669, −7.223132463467104, −6.720066100617865, −5.960520268394544, −5.741323927958606, −4.865171997141630, −4.424406129468090, −3.862361409858416, −3.013036416644023, −2.620974773614891, −2.007285676274223, −0.9262751546285971, 0, 0.9262751546285971, 2.007285676274223, 2.620974773614891, 3.013036416644023, 3.862361409858416, 4.424406129468090, 4.865171997141630, 5.741323927958606, 5.960520268394544, 6.720066100617865, 7.223132463467104, 7.671802963374669, 8.283806951536278, 9.161810285238644, 9.431420879070070, 10.01964282591604, 10.76367957690681, 10.93322720466951, 11.68012595404732, 12.31510876453235, 12.64584504154000, 13.01185763411863, 13.53509834236016, 14.14118717436648, 14.63917718491213

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