# Properties

 Degree 2 Conductor 3 Sign $-0.818 + 0.574i$ Motivic weight 16 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 312. i·2-s + (−5.36e3 + 3.77e3i)3-s − 3.22e4·4-s − 2.76e5i·5-s + (−1.17e6 − 1.67e6i)6-s − 7.10e6·7-s + 1.04e7i·8-s + (1.46e7 − 4.04e7i)9-s + 8.65e7·10-s + 3.43e8i·11-s + (1.73e8 − 1.21e8i)12-s − 7.14e8·13-s − 2.22e9i·14-s + (1.04e9 + 1.48e9i)15-s − 5.36e9·16-s + 6.74e8i·17-s + ⋯
 L(s)  = 1 + 1.22i·2-s + (−0.818 + 0.574i)3-s − 0.492·4-s − 0.708i·5-s + (−0.701 − 0.999i)6-s − 1.23·7-s + 0.620i·8-s + (0.339 − 0.940i)9-s + 0.865·10-s + 1.60i·11-s + (0.402 − 0.282i)12-s − 0.876·13-s − 1.50i·14-s + (0.407 + 0.579i)15-s − 1.24·16-s + 0.0967i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(17-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3$$ $$\varepsilon$$ = $-0.818 + 0.574i$ motivic weight = $$16$$ character : $\chi_{3} (2, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3,\ (\ :8),\ -0.818 + 0.574i)$ $L(\frac{17}{2})$ $\approx$ $0.189391 - 0.599352i$ $L(\frac12)$ $\approx$ $0.189391 - 0.599352i$ $L(9)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 3$, $$F_p$$ is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + (5.36e3 - 3.77e3i)T$$
good2 $$1 - 312. iT - 6.55e4T^{2}$$
5 $$1 + 2.76e5iT - 1.52e11T^{2}$$
7 $$1 + 7.10e6T + 3.32e13T^{2}$$
11 $$1 - 3.43e8iT - 4.59e16T^{2}$$
13 $$1 + 7.14e8T + 6.65e17T^{2}$$
17 $$1 - 6.74e8iT - 4.86e19T^{2}$$
19 $$1 - 4.70e9T + 2.88e20T^{2}$$
23 $$1 - 4.47e10iT - 6.13e21T^{2}$$
29 $$1 - 3.75e11iT - 2.50e23T^{2}$$
31 $$1 - 6.58e11T + 7.27e23T^{2}$$
37 $$1 - 3.89e11T + 1.23e25T^{2}$$
41 $$1 + 4.14e12iT - 6.37e25T^{2}$$
43 $$1 + 1.68e13T + 1.36e26T^{2}$$
47 $$1 - 9.08e12iT - 5.66e26T^{2}$$
53 $$1 - 4.32e13iT - 3.87e27T^{2}$$
59 $$1 + 2.07e14iT - 2.15e28T^{2}$$
61 $$1 - 1.10e14T + 3.67e28T^{2}$$
67 $$1 - 1.61e14T + 1.64e29T^{2}$$
71 $$1 - 8.06e13iT - 4.16e29T^{2}$$
73 $$1 + 8.85e14T + 6.50e29T^{2}$$
79 $$1 - 2.35e15T + 2.30e30T^{2}$$
83 $$1 - 3.93e15iT - 5.07e30T^{2}$$
89 $$1 - 3.56e15iT - 1.54e31T^{2}$$
97 $$1 + 1.51e15T + 6.14e31T^{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

−23.56523187292387208434549521472, −22.43281748559220214795500563929, −20.26672307508967734396696774218, −17.47825548440015866452895385767, −16.38522057107798428245705012450, −15.20664748669160562869486037550, −12.42351768649475129769376715329, −9.641806859839399850193019481738, −6.84783688008767126190920179119, −4.99031573645298447056423255859, 0.42218955733838694584384415923, 2.90750237650572410083716922065, 6.53455730748199774289560544157, 10.27199668072309756602977930882, 11.71164296627906324522090579035, 13.26976432337153163029235510906, 16.37787049749053713916100802538, 18.65617881665796415201514059676, 19.44938708742468372066300814206, 21.77955848001828004173888235131