# Properties

 Degree 2 Conductor $2^{2} \cdot 3$ Sign $0.555 + 0.831i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (−1.73 − 2.23i)2-s + (3.46 − 3.87i)3-s + (−2.00 + 7.74i)4-s + 8.94i·5-s + (−14.6 − 1.03i)6-s + 7.74i·7-s + (20.7 − 8.94i)8-s + (−3.00 − 26.8i)9-s + (20.0 − 15.4i)10-s − 34.6·11-s + (23.0 + 34.5i)12-s − 10·13-s + (17.3 − 13.4i)14-s + (34.6 + 30.9i)15-s + (−56 − 30.9i)16-s + 35.7i·17-s + ⋯
 L(s)  = 1 + (−0.612 − 0.790i)2-s + (0.666 − 0.745i)3-s + (−0.250 + 0.968i)4-s + 0.799i·5-s + (−0.997 − 0.0706i)6-s + 0.418i·7-s + (0.918 − 0.395i)8-s + (−0.111 − 0.993i)9-s + (0.632 − 0.489i)10-s − 0.949·11-s + (0.555 + 0.831i)12-s − 0.213·13-s + (0.330 − 0.256i)14-s + (0.596 + 0.533i)15-s + (−0.875 − 0.484i)16-s + 0.510i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$12$$    =    $$2^{2} \cdot 3$$ $$\varepsilon$$ = $0.555 + 0.831i$ motivic weight = $$3$$ character : $\chi_{12} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 12,\ (\ :3/2),\ 0.555 + 0.831i)$ $L(2)$ $\approx$ $0.710465 - 0.380053i$ $L(\frac12)$ $\approx$ $0.710465 - 0.380053i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + (1.73 + 2.23i)T$$
3 $$1 + (-3.46 + 3.87i)T$$
good5 $$1 - 8.94iT - 125T^{2}$$
7 $$1 - 7.74iT - 343T^{2}$$
11 $$1 + 34.6T + 1.33e3T^{2}$$
13 $$1 + 10T + 2.19e3T^{2}$$
17 $$1 - 35.7iT - 4.91e3T^{2}$$
19 $$1 + 69.7iT - 6.85e3T^{2}$$
23 $$1 - 96.9T + 1.21e4T^{2}$$
29 $$1 + 152. iT - 2.43e4T^{2}$$
31 $$1 - 224. iT - 2.97e4T^{2}$$
37 $$1 + 130T + 5.06e4T^{2}$$
41 $$1 + 125. iT - 6.89e4T^{2}$$
43 $$1 + 224. iT - 7.95e4T^{2}$$
47 $$1 - 193.T + 1.03e5T^{2}$$
53 $$1 - 545. iT - 1.48e5T^{2}$$
59 $$1 + 173.T + 2.05e5T^{2}$$
61 $$1 + 442T + 2.26e5T^{2}$$
67 $$1 - 735. iT - 3.00e5T^{2}$$
71 $$1 + 1.03e3T + 3.57e5T^{2}$$
73 $$1 - 410T + 3.89e5T^{2}$$
79 $$1 + 85.2iT - 4.93e5T^{2}$$
83 $$1 - 1.25e3T + 5.71e5T^{2}$$
89 $$1 - 840. iT - 7.04e5T^{2}$$
97 $$1 - 770T + 9.12e5T^{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

−19.31859888442374478632059986411, −18.57519266735791446315068748718, −17.51155180117663029412240145959, −15.30865368685834500042340144278, −13.61236202681288569750388567473, −12.30827082394096717786522346978, −10.63583717773865906397103834563, −8.826431911939698895359179639394, −7.25318987538709189929739072122, −2.75108982179113338574342640129, 4.99667685049821096552388374399, 7.79450003352070673570454316107, 9.171279628502259032815510915885, 10.53836900377380684539559835725, 13.33606714947912092173490198699, 14.75833816652542838209323584236, 16.05181287365442182297584093573, 16.89791796292965165676381253160, 18.64062128153949712611853776322, 20.04615591368134888172286197739