Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \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 − 2·5-s + 7-s − 3·8-s − 2·10-s + 4·11-s − 2·13-s + 14-s − 16-s + 4·19-s + 2·20-s + 4·22-s − 25-s − 2·26-s − 28-s − 2·29-s + 5·32-s − 2·35-s − 6·37-s + 4·38-s + 6·40-s + 2·41-s − 4·43-s − 4·44-s + 49-s − 50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s − 1.06·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s + 0.883·32-s − 0.338·35-s − 0.986·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s − 0.603·44-s + 1/7·49-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(18207\)    =    \(3^{2} \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{18207} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 18207,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.95066628362837, −15.23694510896951, −14.99347282634105, −14.36447229976918, −13.87339294285049, −13.53695528219071, −12.56693599249134, −12.24393215700795, −11.79650828416118, −11.32659985566129, −10.62940818322475, −9.673888266405059, −9.367959610241320, −8.745853962455529, −8.013096519700251, −7.587425002085384, −6.775282039674580, −6.209136998053063, −5.346370306353391, −4.876916872726395, −4.229666405576952, −3.615704252971583, −3.214406595570456, −2.060674137441135, −1.028751621894537, 0, 1.028751621894537, 2.060674137441135, 3.214406595570456, 3.615704252971583, 4.229666405576952, 4.876916872726395, 5.346370306353391, 6.209136998053063, 6.775282039674580, 7.587425002085384, 8.013096519700251, 8.745853962455529, 9.367959610241320, 9.673888266405059, 10.62940818322475, 11.32659985566129, 11.79650828416118, 12.24393215700795, 12.56693599249134, 13.53695528219071, 13.87339294285049, 14.36447229976918, 14.99347282634105, 15.23694510896951, 15.95066628362837

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