Properties

Degree 2
Conductor $ 3 \cdot 5 \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 − 3-s − 4-s − 5-s + 6-s − 7-s + 3·8-s + 9-s + 10-s + 4·11-s + 12-s − 2·13-s + 14-s + 15-s − 16-s − 18-s − 4·19-s + 20-s + 21-s − 4·22-s + 8·23-s − 3·24-s + 25-s + 2·26-s − 27-s + 28-s − 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30345\)    =    \(3 \cdot 5 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30345} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 30345,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
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

−15.32837471660586, −14.81340284367589, −14.58481016069975, −13.69561633302872, −13.14669440370638, −12.73674082325287, −12.21406929076868, −11.52378790380297, −10.99401144680784, −10.68748591545752, −9.788839558999460, −9.474787694715793, −8.982126148370341, −8.450370024382365, −7.678763031474116, −7.206952169185408, −6.679658029176299, −6.003152195067085, −5.288930217634256, −4.525218213333510, −4.196021758510845, −3.482861708759338, −2.537784841001403, −1.502154858266484, −0.8349867982987431, 0, 0.8349867982987431, 1.502154858266484, 2.537784841001403, 3.482861708759338, 4.196021758510845, 4.525218213333510, 5.288930217634256, 6.003152195067085, 6.679658029176299, 7.206952169185408, 7.678763031474116, 8.450370024382365, 8.982126148370341, 9.474787694715793, 9.788839558999460, 10.68748591545752, 10.99401144680784, 11.52378790380297, 12.21406929076868, 12.73674082325287, 13.14669440370638, 13.69561633302872, 14.58481016069975, 14.81340284367589, 15.32837471660586

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