# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{4} \cdot 7$ Sign $-0.173 - 0.984i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)7-s + (3 + 5.19i)11-s + (−1 + 1.73i)13-s − 4·19-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + (−3 − 5.19i)29-s + (−4 + 6.92i)31-s + 2·37-s + (−6 + 10.3i)41-s + (2 + 3.46i)43-s + (−6 − 10.3i)47-s + (−0.499 + 0.866i)49-s − 6·53-s + (5 + 8.66i)61-s + ⋯
 L(s)  = 1 + (−0.188 − 0.327i)7-s + (0.904 + 1.56i)11-s + (−0.277 + 0.480i)13-s − 0.917·19-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + (−0.557 − 0.964i)29-s + (−0.718 + 1.24i)31-s + 0.328·37-s + (−0.937 + 1.62i)41-s + (0.304 + 0.528i)43-s + (−0.875 − 1.51i)47-s + (−0.0714 + 0.123i)49-s − 0.824·53-s + (0.640 + 1.10i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2268$$    =    $$2^{2} \cdot 3^{4} \cdot 7$$ $$\varepsilon$$ = $-0.173 - 0.984i$ motivic weight = $$1$$ character : $\chi_{2268} (757, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 2268,\ (\ :1/2),\ -0.173 - 0.984i)$$ $$L(1)$$ $$\approx$$ $$1.264580172$$ $$L(\frac12)$$ $$\approx$$ $$1.264580172$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1 + (0.5 + 0.866i)T$$
good5 $$1 + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 + 4T + 19T^{2}$$
23 $$1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 - 2T + 37T^{2}$$
41 $$1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 6T + 71T^{2}$$
73 $$1 + 10T + 73T^{2}$$
79 $$1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 12T + 89T^{2}$$
97 $$1 + (-5 - 8.66i)T + (-48.5 + 84.0i)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

−9.277514125400977374924380205898, −8.581633392470210362959985492589, −7.55487425682306110698043040046, −6.80280541089586687415480976047, −6.45421799154760728205789787926, −5.04670819917669227594888150587, −4.46884447186603265294106039120, −3.61902159725980669399051113580, −2.36346957374264415656614906676, −1.39522684320367510089473238635, 0.43932082541865337390556094970, 1.83537165811994897304706491537, 3.11662560351249626213377973979, 3.71241073276804783745447111211, 4.87651503975233622159056832727, 5.80930879213329643422588999269, 6.31189553496052650422038856896, 7.26718065801961995198100705756, 8.139114171419124658855805662152, 8.916673708942122618435374206025