Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 2·5-s + 6-s + 7-s + 3·8-s + 9-s − 2·10-s − 4·11-s + 12-s − 2·13-s − 14-s − 2·15-s − 16-s − 18-s + 4·19-s − 2·20-s − 21-s + 4·22-s − 3·24-s − 25-s + 2·26-s − 27-s − 28-s + 2·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s − 0.612·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6069\)    =    \(3 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6069} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6069,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8752924544$
$L(\frac12)$  $\approx$  $0.8752924544$
$L(\frac{3}{2})$   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 \( 1 + T \)
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

−17.63266736995595, −16.89272869233006, −16.56699002618150, −15.72800611202897, −15.21094327555162, −14.15807926329448, −13.88992089602343, −13.19775078753438, −12.70547863504290, −11.89047697447887, −11.18621308143610, −10.43392362099404, −9.998513844079506, −9.645288120064412, −8.729407214939294, −8.150173736744292, −7.432760897506093, −6.836520659252350, −5.706356007139538, −5.244635712026632, −4.776303424819420, −3.690508959525003, −2.482320802332792, −1.652351551486228, −0.5780766878793333, 0.5780766878793333, 1.652351551486228, 2.482320802332792, 3.690508959525003, 4.776303424819420, 5.244635712026632, 5.706356007139538, 6.836520659252350, 7.432760897506093, 8.150173736744292, 8.729407214939294, 9.645288120064412, 9.998513844079506, 10.43392362099404, 11.18621308143610, 11.89047697447887, 12.70547863504290, 13.19775078753438, 13.88992089602343, 14.15807926329448, 15.21094327555162, 15.72800611202897, 16.56699002618150, 16.89272869233006, 17.63266736995595

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