# Properties

 Label 2-690-3.2-c2-0-25 Degree $2$ Conductor $690$ Sign $-0.847 - 0.531i$ Analytic cond. $18.8011$ Root an. cond. $4.33602$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 1.41i·2-s + (−1.59 + 2.54i)3-s − 2.00·4-s + 2.23i·5-s + (−3.59 − 2.25i)6-s + 10.3·7-s − 2.82i·8-s + (−3.92 − 8.10i)9-s − 3.16·10-s + 14.4i·11-s + (3.18 − 5.08i)12-s + 14.5·13-s + 14.5i·14-s + (−5.68 − 3.56i)15-s + 4.00·16-s − 0.438i·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s + (−0.531 + 0.847i)3-s − 0.500·4-s + 0.447i·5-s + (−0.599 − 0.375i)6-s + 1.47·7-s − 0.353i·8-s + (−0.435 − 0.900i)9-s − 0.316·10-s + 1.31i·11-s + (0.265 − 0.423i)12-s + 1.12·13-s + 1.04i·14-s + (−0.378 − 0.237i)15-s + 0.250·16-s − 0.0257i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$690$$    =    $$2 \cdot 3 \cdot 5 \cdot 23$$ Sign: $-0.847 - 0.531i$ Analytic conductor: $$18.8011$$ Root analytic conductor: $$4.33602$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{690} (461, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 690,\ (\ :1),\ -0.847 - 0.531i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.792124478$$ $$L(\frac12)$$ $$\approx$$ $$1.792124478$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 1.41iT$$
3 $$1 + (1.59 - 2.54i)T$$
5 $$1 - 2.23iT$$
23 $$1 - 4.79iT$$
good7 $$1 - 10.3T + 49T^{2}$$
11 $$1 - 14.4iT - 121T^{2}$$
13 $$1 - 14.5T + 169T^{2}$$
17 $$1 + 0.438iT - 289T^{2}$$
19 $$1 - 30.5T + 361T^{2}$$
29 $$1 - 23.6iT - 841T^{2}$$
31 $$1 - 37.8T + 961T^{2}$$
37 $$1 - 3.95T + 1.36e3T^{2}$$
41 $$1 - 0.897iT - 1.68e3T^{2}$$
43 $$1 + 47.5T + 1.84e3T^{2}$$
47 $$1 - 39.9iT - 2.20e3T^{2}$$
53 $$1 + 67.0iT - 2.80e3T^{2}$$
59 $$1 + 43.7iT - 3.48e3T^{2}$$
61 $$1 + 69.1T + 3.72e3T^{2}$$
67 $$1 - 15.9T + 4.48e3T^{2}$$
71 $$1 + 32.7iT - 5.04e3T^{2}$$
73 $$1 + 136.T + 5.32e3T^{2}$$
79 $$1 + 117.T + 6.24e3T^{2}$$
83 $$1 + 72.3iT - 6.88e3T^{2}$$
89 $$1 - 48.7iT - 7.92e3T^{2}$$
97 $$1 - 128.T + 9.40e3T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.51770018917795748072523007883, −9.808622610639547791350419520510, −8.873890229163253143158504999958, −7.949627685727703295206141872659, −7.10667768309471226534686876400, −6.07446308911607043619693946076, −5.06951367354411170527030383186, −4.54913812728969085740720495907, −3.34514085138136966657764883527, −1.39029547423532614776486635058, 0.846146480351116671107964335104, 1.52115990028203470654015271232, 2.99719281704540928740028301815, 4.41938810315252604306979724778, 5.39997453081499335314333070601, 6.06732450401704904441998495345, 7.52847832409368008103111618806, 8.310862652723513540310781804363, 8.763171199405511290173670029691, 10.20981925719390202031335381558