# Properties

 Label 2-69-23.22-c6-0-19 Degree $2$ Conductor $69$ Sign $0.493 + 0.869i$ Analytic cond. $15.8737$ Root an. cond. $3.98418$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 13.3·2-s − 15.5·3-s + 114.·4-s − 40.7i·5-s − 208.·6-s − 469. i·7-s + 675.·8-s + 243·9-s − 544. i·10-s − 1.56e3i·11-s − 1.78e3·12-s + 3.79e3·13-s − 6.27e3i·14-s + 635. i·15-s + 1.69e3·16-s + 9.02e3i·17-s + ⋯
 L(s)  = 1 + 1.67·2-s − 0.577·3-s + 1.78·4-s − 0.326i·5-s − 0.964·6-s − 1.36i·7-s + 1.31·8-s + 0.333·9-s − 0.544i·10-s − 1.17i·11-s − 1.03·12-s + 1.72·13-s − 2.28i·14-s + 0.188i·15-s + 0.413·16-s + 1.83i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 + 0.869i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$69$$    =    $$3 \cdot 23$$ Sign: $0.493 + 0.869i$ Analytic conductor: $$15.8737$$ Root analytic conductor: $$3.98418$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{69} (22, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 69,\ (\ :3),\ 0.493 + 0.869i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$3.44256 - 2.00551i$$ $$L(\frac12)$$ $$\approx$$ $$3.44256 - 2.00551i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 15.5T$$
23 $$1 + (6.00e3 + 1.05e4i)T$$
good2 $$1 - 13.3T + 64T^{2}$$
5 $$1 + 40.7iT - 1.56e4T^{2}$$
7 $$1 + 469. iT - 1.17e5T^{2}$$
11 $$1 + 1.56e3iT - 1.77e6T^{2}$$
13 $$1 - 3.79e3T + 4.82e6T^{2}$$
17 $$1 - 9.02e3iT - 2.41e7T^{2}$$
19 $$1 + 7.10e3iT - 4.70e7T^{2}$$
29 $$1 + 3.65e3T + 5.94e8T^{2}$$
31 $$1 + 2.75e4T + 8.87e8T^{2}$$
37 $$1 - 7.51e4iT - 2.56e9T^{2}$$
41 $$1 + 1.16e3T + 4.75e9T^{2}$$
43 $$1 - 7.70e4iT - 6.32e9T^{2}$$
47 $$1 - 1.28e5T + 1.07e10T^{2}$$
53 $$1 - 1.49e5iT - 2.21e10T^{2}$$
59 $$1 + 5.59e4T + 4.21e10T^{2}$$
61 $$1 + 1.67e5iT - 5.15e10T^{2}$$
67 $$1 - 1.99e5iT - 9.04e10T^{2}$$
71 $$1 - 1.20e5T + 1.28e11T^{2}$$
73 $$1 - 6.59e5T + 1.51e11T^{2}$$
79 $$1 - 4.58e5iT - 2.43e11T^{2}$$
83 $$1 - 2.93e5iT - 3.26e11T^{2}$$
89 $$1 + 4.30e4iT - 4.96e11T^{2}$$
97 $$1 + 8.56e5iT - 8.32e11T^{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

−13.35152465184363432021123549269, −12.62643835188365202575382421656, −11.04040364675975881763374935890, −10.81190249253243351007543711143, −8.436767992226026046500850750239, −6.66061642999113385954422082868, −5.88324500626046990538626125985, −4.39155120708257403181402409360, −3.52517052029187335447792193180, −1.07131371376534420386286105074, 2.12557648180179629227571092154, 3.66714432920225969930290966284, 5.18946320133748025175708179255, 5.95293428725246473208849145275, 7.18618487164893973366640314838, 9.213126536517667681675381987691, 10.92720895233970921909925710536, 11.89015760192043964355339228446, 12.50622632437103251920455526416, 13.65506581632664842412341738261