# Properties

 Label 2-690-115.22-c1-0-23 Degree $2$ Conductor $690$ Sign $0.999 - 0.0149i$ Analytic cond. $5.50967$ Root an. cond. $2.34727$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + (2.19 + 0.408i)5-s + 1.00·6-s + (1.64 − 1.64i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (1.26 + 1.84i)10-s − 6.11i·11-s + (0.707 + 0.707i)12-s + (−0.241 + 0.241i)13-s + 2.32·14-s + (1.84 − 1.26i)15-s − 1.00·16-s + (0.327 − 0.327i)17-s + ⋯
 L(s)  = 1 + (0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + (0.983 + 0.182i)5-s + 0.408·6-s + (0.622 − 0.622i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.400 + 0.582i)10-s − 1.84i·11-s + (0.204 + 0.204i)12-s + (−0.0669 + 0.0669i)13-s + 0.622·14-s + (0.475 − 0.326i)15-s − 0.250·16-s + (0.0793 − 0.0793i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0149i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0149i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$690$$    =    $$2 \cdot 3 \cdot 5 \cdot 23$$ Sign: $0.999 - 0.0149i$ Analytic conductor: $$5.50967$$ Root analytic conductor: $$2.34727$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{690} (367, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 690,\ (\ :1/2),\ 0.999 - 0.0149i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.69896 + 0.0201666i$$ $$L(\frac12)$$ $$\approx$$ $$2.69896 + 0.0201666i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.707 - 0.707i)T$$
3 $$1 + (-0.707 + 0.707i)T$$
5 $$1 + (-2.19 - 0.408i)T$$
23 $$1 + (1.66 - 4.49i)T$$
good7 $$1 + (-1.64 + 1.64i)T - 7iT^{2}$$
11 $$1 + 6.11iT - 11T^{2}$$
13 $$1 + (0.241 - 0.241i)T - 13iT^{2}$$
17 $$1 + (-0.327 + 0.327i)T - 17iT^{2}$$
19 $$1 + 0.759T + 19T^{2}$$
29 $$1 - 7.09iT - 29T^{2}$$
31 $$1 + 8.30T + 31T^{2}$$
37 $$1 + (1.92 - 1.92i)T - 37iT^{2}$$
41 $$1 - 6.75T + 41T^{2}$$
43 $$1 + (0.144 + 0.144i)T + 43iT^{2}$$
47 $$1 + (-1.46 - 1.46i)T + 47iT^{2}$$
53 $$1 + (-1.45 - 1.45i)T + 53iT^{2}$$
59 $$1 - 1.68iT - 59T^{2}$$
61 $$1 - 8.05iT - 61T^{2}$$
67 $$1 + (3.13 - 3.13i)T - 67iT^{2}$$
71 $$1 - 6.24T + 71T^{2}$$
73 $$1 + (-6.62 + 6.62i)T - 73iT^{2}$$
79 $$1 + 3.83T + 79T^{2}$$
83 $$1 + (2.14 + 2.14i)T + 83iT^{2}$$
89 $$1 - 4.64T + 89T^{2}$$
97 $$1 + (11.2 - 11.2i)T - 97iT^{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

−10.69505197143455948801674692651, −9.353484036932780025297798091633, −8.660802370752582367012846117030, −7.74172213263063488025007543789, −6.92446347963341951531511264760, −5.92561087744827331611401093144, −5.30868122006706986860350415825, −3.85559732035412480207850259653, −2.88522347028771703272920014135, −1.40378769734152605089926593791, 1.91887709586518433604909655907, 2.40605097585882379141630197189, 4.07709093101314337509141116277, 4.89424034967589281790462050406, 5.66143318008967579258333727802, 6.81382681399499354893798743803, 7.988464080577670195072681105347, 9.069268799151571885700044829976, 9.693758440646242457744213166646, 10.34215826457343462067010110115