# Properties

 Degree 2 Conductor $2 \cdot 7 \cdot 11$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 2·3-s + 4-s + 2·5-s − 2·6-s − 7-s − 8-s + 9-s − 2·10-s + 11-s + 2·12-s − 4·13-s + 14-s + 4·15-s + 16-s − 18-s + 4·19-s + 2·20-s − 2·21-s − 22-s + 4·23-s − 2·24-s − 25-s + 4·26-s − 4·27-s − 28-s + 2·29-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.10·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.436·21-s − 0.213·22-s + 0.834·23-s − 0.408·24-s − 1/5·25-s + 0.784·26-s − 0.769·27-s − 0.188·28-s + 0.371·29-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$154$$    =    $$2 \cdot 7 \cdot 11$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{154} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 154,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.219984286$ $L(\frac12)$ $\approx$ $1.219984286$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;7,\;11\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + T$$
7 $$1 + T$$
11 $$1 - T$$
good3 $$1 - 2 T + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 + 10 T + p T^{2}$$
37 $$1 + 6 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 - 10 T + p T^{2}$$
53 $$1 + 14 T + p T^{2}$$
59 $$1 - 10 T + p T^{2}$$
61 $$1 + 8 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 + 4 T + p T^{2}$$
73 $$1 - 4 T + p T^{2}$$
79 $$1 - 16 T + p T^{2}$$
83 $$1 - 4 T + p T^{2}$$
89 $$1 - 10 T + p T^{2}$$
97 $$1 - 6 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

−19.39656864364458, −18.59495098478940, −17.52706975746637, −16.90034323595959, −15.75764963262840, −14.72798367784496, −14.07327022990089, −13.12237494463203, −11.98890555673220, −10.59710461621408, −9.423423700744115, −9.240853610425597, −7.873596397691486, −6.890704282695376, −5.417455389505031, −3.299662816513440, −2.069197054453079, 2.069197054453079, 3.299662816513440, 5.417455389505031, 6.890704282695376, 7.873596397691486, 9.240853610425597, 9.423423700744115, 10.59710461621408, 11.98890555673220, 13.12237494463203, 14.07327022990089, 14.72798367784496, 15.75764963262840, 16.90034323595959, 17.52706975746637, 18.59495098478940, 19.39656864364458