Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $0.310 - 0.950i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (4.26 + 2.60i)5-s − 2.44·6-s + (1.87 − 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + (−6.87 + 1.66i)10-s + 11.5·11-s + (2.44 − 2.44i)12-s + (−2.01 − 2.01i)13-s + 3.74i·14-s + (2.04 + 8.41i)15-s − 4·16-s + (7.75 − 7.75i)17-s + ⋯
 L(s)  = 1 + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (0.853 + 0.520i)5-s − 0.408·6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.687 + 0.166i)10-s + 1.05·11-s + (0.204 − 0.204i)12-s + (−0.155 − 0.155i)13-s + 0.267i·14-s + (0.136 + 0.561i)15-s − 0.250·16-s + (0.456 − 0.456i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.310 - 0.950i$ motivic weight = $$2$$ character : $\chi_{210} (43, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1),\ 0.310 - 0.950i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$1.33438 + 0.967871i$$ $$L(\frac12)$$ $$\approx$$ $$1.33438 + 0.967871i$$ $$L(2)$$ 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 + (1 - i)T$$
3 $$1 + (-1.22 - 1.22i)T$$
5 $$1 + (-4.26 - 2.60i)T$$
7 $$1 + (-1.87 + 1.87i)T$$
good11 $$1 - 11.5T + 121T^{2}$$
13 $$1 + (2.01 + 2.01i)T + 169iT^{2}$$
17 $$1 + (-7.75 + 7.75i)T - 289iT^{2}$$
19 $$1 - 21.8iT - 361T^{2}$$
23 $$1 + (7.41 + 7.41i)T + 529iT^{2}$$
29 $$1 - 31.0iT - 841T^{2}$$
31 $$1 + 11.5T + 961T^{2}$$
37 $$1 + (32.0 - 32.0i)T - 1.36e3iT^{2}$$
41 $$1 - 39.6T + 1.68e3T^{2}$$
43 $$1 + (19.3 + 19.3i)T + 1.84e3iT^{2}$$
47 $$1 + (-21.8 + 21.8i)T - 2.20e3iT^{2}$$
53 $$1 + (42.4 + 42.4i)T + 2.80e3iT^{2}$$
59 $$1 + 89.0iT - 3.48e3T^{2}$$
61 $$1 - 7.07T + 3.72e3T^{2}$$
67 $$1 + (-15.6 + 15.6i)T - 4.48e3iT^{2}$$
71 $$1 - 133.T + 5.04e3T^{2}$$
73 $$1 + (92.3 + 92.3i)T + 5.32e3iT^{2}$$
79 $$1 + 126. iT - 6.24e3T^{2}$$
83 $$1 + (30.0 + 30.0i)T + 6.88e3iT^{2}$$
89 $$1 + 4.93iT - 7.92e3T^{2}$$
97 $$1 + (30.2 - 30.2i)T - 9.40e3iT^{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

−12.30439108405937700428930487154, −11.01822163225181339095987321589, −10.12951379945265843272954847706, −9.454777448598201781730165178436, −8.444361509552160388712121343659, −7.27992493614826057610857360367, −6.27675981576519263789582057752, −5.09761096536803809239693154799, −3.49068652235121246833666757195, −1.71009064446271767592586966835, 1.25736055090861230815544012856, 2.47425528455941239649156362507, 4.17344231096642841782680404335, 5.74598428652444534795028695545, 6.96032737694361188736390580007, 8.219009342589847556362326158635, 9.133910154896714984421447693786, 9.682913300740965758027353573373, 11.00170578860315062966478164106, 12.02467894596991120831636432827