Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 17 $
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 − 8-s + 9-s − 10-s + 11-s + 12-s + 2·13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 4·19-s + 20-s + 21-s − 22-s − 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 − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.213·22-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 39270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 39270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(39270\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{39270} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 39270,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;11,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
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

−15.03071548735733, −14.60678385180776, −14.03302348715042, −13.66776912842595, −12.86816790965363, −12.61215412707311, −11.84564383063097, −11.22585278057930, −10.84167258728862, −10.18046277783600, −9.825229204955544, −9.069439468972885, −8.675664545943381, −8.356273490951712, −7.616055175972977, −7.075028278125713, −6.452754048257517, −5.977573346815168, −5.239868859950243, −4.397202309275230, −3.944661945057742, −2.986778040185100, −2.505513967121129, −1.654554382891277, −1.221565515433384, 0, 1.221565515433384, 1.654554382891277, 2.505513967121129, 2.986778040185100, 3.944661945057742, 4.397202309275230, 5.239868859950243, 5.977573346815168, 6.452754048257517, 7.075028278125713, 7.616055175972977, 8.356273490951712, 8.675664545943381, 9.069439468972885, 9.825229204955544, 10.18046277783600, 10.84167258728862, 11.22585278057930, 11.84564383063097, 12.61215412707311, 12.86816790965363, 13.66776912842595, 14.03302348715042, 14.60678385180776, 15.03071548735733

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