# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $0.999 + 0.00120i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 + 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s − 2.44i·6-s + (4.24 − 5.56i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (2.73 + 1.58i)10-s + (5.42 − 9.40i)11-s + (2.99 − 1.73i)12-s + 0.772i·13-s + (9.81 + 1.26i)14-s − 3.87·15-s + (−2.00 − 3.46i)16-s + (16.7 + 9.68i)17-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s − 0.408i·6-s + (0.606 − 0.795i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.273 + 0.158i)10-s + (0.493 − 0.854i)11-s + (0.249 − 0.144i)12-s + 0.0593i·13-s + (0.701 + 0.0902i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.986 + 0.569i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00120i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00120i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.999 + 0.00120i$ motivic weight = $$2$$ character : $\chi_{210} (31, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1),\ 0.999 + 0.00120i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.76957 - 0.00106236i$$ $$L(\frac12)$$ $$\approx$$ $$1.76957 - 0.00106236i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.707 - 1.22i)T$$
3 $$1 + (1.5 + 0.866i)T$$
5 $$1 + (-1.93 + 1.11i)T$$
7 $$1 + (-4.24 + 5.56i)T$$
good11 $$1 + (-5.42 + 9.40i)T + (-60.5 - 104. i)T^{2}$$
13 $$1 - 0.772iT - 169T^{2}$$
17 $$1 + (-16.7 - 9.68i)T + (144.5 + 250. i)T^{2}$$
19 $$1 + (-22.5 + 13.0i)T + (180.5 - 312. i)T^{2}$$
23 $$1 + (-6.84 - 11.8i)T + (-264.5 + 458. i)T^{2}$$
29 $$1 + 6.99T + 841T^{2}$$
31 $$1 + (-22.7 - 13.1i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + (32.3 + 55.9i)T + (-684.5 + 1.18e3i)T^{2}$$
41 $$1 + 5.54iT - 1.68e3T^{2}$$
43 $$1 + 68.9T + 1.84e3T^{2}$$
47 $$1 + (19.5 - 11.3i)T + (1.10e3 - 1.91e3i)T^{2}$$
53 $$1 + (37.2 - 64.5i)T + (-1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (-96.6 - 55.8i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (46.9 - 27.1i)T + (1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-22.1 + 38.3i)T + (-2.24e3 - 3.88e3i)T^{2}$$
71 $$1 - 31.9T + 5.04e3T^{2}$$
73 $$1 + (92.6 + 53.4i)T + (2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (14.8 + 25.7i)T + (-3.12e3 + 5.40e3i)T^{2}$$
83 $$1 - 15.8iT - 6.88e3T^{2}$$
89 $$1 + (31.8 - 18.3i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 - 134. iT - 9.40e3T^{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

−12.16121538786538754856410874346, −11.34144745778724143323375563773, −10.27998405640106200238493723375, −9.016516862853258934331787772552, −7.87740357138089137737258910959, −6.98613285381387366560009887803, −5.82798294801867947246358509485, −4.95293590409190754437598790517, −3.52910450614419869575977570498, −1.15902809124987229929026598409, 1.58976726596958184663327864551, 3.19363976635606995269604861580, 4.79104181325984302140693570932, 5.53743677976992720501840635130, 6.80010788613765180166400658380, 8.286392493409147733149985807021, 9.681914781975932205455140365994, 10.07070529735011587306228716315, 11.54940686394528545639110170556, 11.84594762059159049571916895603