# Properties

 Degree 2 Conductor $3 \cdot 37$ Sign $0.621 - 0.783i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + (−1 + 1.73i)19-s + (0.499 + 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.499 + 0.866i)28-s − 31-s + 0.999·36-s + 37-s + ⋯
 L(s)  = 1 + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + (−1 + 1.73i)19-s + (0.499 + 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (0.499 + 0.866i)28-s − 31-s + 0.999·36-s + 37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$111$$    =    $$3 \cdot 37$$ $$\varepsilon$$ = $0.621 - 0.783i$ motivic weight = $$0$$ character : $\chi_{111} (26, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 111,\ (\ :0),\ 0.621 - 0.783i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.4872960951$$ $$L(\frac12)$$ $$\approx$$ $$0.4872960951$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;37\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (0.5 - 0.866i)T$$
37 $$1 - T$$
good2 $$1 + (0.5 - 0.866i)T^{2}$$
5 $$1 + (0.5 + 0.866i)T^{2}$$
7 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
11 $$1 - T^{2}$$
13 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
17 $$1 + (0.5 - 0.866i)T^{2}$$
19 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}$$
23 $$1 - T^{2}$$
29 $$1 - T^{2}$$
31 $$1 + T + T^{2}$$
41 $$1 + (0.5 + 0.866i)T^{2}$$
43 $$1 + T + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (0.5 - 0.866i)T^{2}$$
59 $$1 + (0.5 - 0.866i)T^{2}$$
61 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
71 $$1 + (0.5 + 0.866i)T^{2}$$
73 $$1 + T + T^{2}$$
79 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
83 $$1 + (0.5 - 0.866i)T^{2}$$
89 $$1 + (0.5 - 0.866i)T^{2}$$
97 $$1 + T + 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

−14.08859395282092172934470765459, −12.93352522765451202314856167685, −11.95902905251912013524634114598, −10.80389228668928613087785431902, −10.00763679155022500534844766937, −8.616652078435595282895870814145, −7.70723203883669501750919805383, −5.97703246902699662443433916656, −4.48691441166618421750396295135, −3.60018671529121378340731000668, 1.94869000785363089009706569365, 4.74283713726553977917887541470, 5.81220143111909919666559615863, 6.86328204660327443830310453966, 8.463172696586380663441779432537, 9.323550309234632306437585417334, 10.96748126909624518403723367049, 11.50359384581484360434866422988, 12.87351548979488832581956316560, 13.62304570253849326090067990079