Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s + 6·13-s + 16-s + 17-s − 4·19-s − 20-s + 22-s − 2·23-s + 25-s − 6·26-s − 10·29-s + 4·31-s − 32-s − 34-s + 10·37-s + 4·38-s + 40-s + 6·41-s + 8·43-s − 44-s + 2·46-s − 4·47-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 1.66·13-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.223·20-s + 0.213·22-s − 0.417·23-s + 1/5·25-s − 1.17·26-s − 1.85·29-s + 0.718·31-s − 0.176·32-s − 0.171·34-s + 1.64·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + 1.21·43-s − 0.150·44-s + 0.294·46-s − 0.583·47-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$16830$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{16830} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 16830,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.277640466$ $L(\frac12)$ $\approx$ $1.277640466$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + T$$
3 $$1$$
5 $$1 + T$$
11 $$1 + T$$
17 $$1 - T$$
good7 $$1 + p T^{2}$$
13 $$1 - 6 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 + 2 T + p T^{2}$$
29 $$1 + 10 T + p T^{2}$$
31 $$1 - 4 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 + 4 T + p T^{2}$$
53 $$1 - 10 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 - 2 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 + 14 T + p T^{2}$$
83 $$1 - 8 T + p T^{2}$$
89 $$1 + 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

−16.10101356419920, −15.42441091069542, −14.91481295301469, −14.43005453242050, −13.57308047905049, −13.04850553246251, −12.65026532529946, −11.79240919835904, −11.22980833125470, −10.94538212528928, −10.33507320811080, −9.559767281586165, −9.106316508519990, −8.319087947454565, −8.081178272430855, −7.385836205759872, −6.686846185422621, −5.929100888038034, −5.665752738071024, −4.376586579920968, −3.975517052620507, −3.143667953897622, −2.321281314012520, −1.436351986986149, −0.5663384447509973, 0.5663384447509973, 1.436351986986149, 2.321281314012520, 3.143667953897622, 3.975517052620507, 4.376586579920968, 5.665752738071024, 5.929100888038034, 6.686846185422621, 7.385836205759872, 8.081178272430855, 8.319087947454565, 9.106316508519990, 9.559767281586165, 10.33507320811080, 10.94538212528928, 11.22980833125470, 11.79240919835904, 12.65026532529946, 13.04850553246251, 13.57308047905049, 14.43005453242050, 14.91481295301469, 15.42441091069542, 16.10101356419920