# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 4·11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s + 4·19-s − 20-s − 21-s + 4·22-s − 4·23-s + 24-s + 25-s − 26-s − 27-s + 28-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 − 1.20·11-s − 0.288·12-s + 0.277·13-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 + 0.852·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$46410$$    =    $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{46410} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 46410,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.536378721$ $L(\frac12)$ $\approx$ $1.536378721$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;17\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5,\;7,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + T$$
5 $$1 + T$$
7 $$1 - T$$
13 $$1 - T$$
17 $$1 - T$$
good11 $$1 + 4 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 - 8 T + 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 - 4 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 - 12 T + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 - 14 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 - 16 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 - 14 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

−14.62163434906689, −14.23481754337526, −13.32849894971997, −13.20611171075597, −12.25072154798458, −11.94461010532115, −11.51454936814350, −10.91803432625503, −10.46950009209881, −9.908850066514771, −9.614672709726173, −8.625394990542098, −8.221791023630370, −7.795597849705295, −7.313534794081126, −6.621739583443339, −6.017510098700992, −5.464022019035269, −4.854576098423762, −4.253728694860611, −3.468355878848701, −2.651161937401499, −2.171406073592287, −0.9225720442748944, −0.6959885242002541, 0.6959885242002541, 0.9225720442748944, 2.171406073592287, 2.651161937401499, 3.468355878848701, 4.253728694860611, 4.854576098423762, 5.464022019035269, 6.017510098700992, 6.621739583443339, 7.313534794081126, 7.795597849705295, 8.221791023630370, 8.625394990542098, 9.614672709726173, 9.908850066514771, 10.46950009209881, 10.91803432625503, 11.51454936814350, 11.94461010532115, 12.25072154798458, 13.20611171075597, 13.32849894971997, 14.23481754337526, 14.62163434906689