# Properties

 Label 2-69-69.68-c5-0-19 Degree $2$ Conductor $69$ Sign $0.801 + 0.598i$ Analytic cond. $11.0664$ Root an. cond. $3.32663$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 1.57i·2-s + (−12.8 − 8.75i)3-s + 29.5·4-s + 37.1·5-s + (−13.7 + 20.2i)6-s + 122. i·7-s − 96.6i·8-s + (89.6 + 225. i)9-s − 58.4i·10-s + 321.·11-s + (−380. − 258. i)12-s − 207.·13-s + 192.·14-s + (−479. − 325. i)15-s + 793.·16-s + 1.64e3·17-s + ⋯
 L(s)  = 1 − 0.277i·2-s + (−0.827 − 0.561i)3-s + 0.922·4-s + 0.665·5-s + (−0.156 + 0.229i)6-s + 0.942i·7-s − 0.534i·8-s + (0.368 + 0.929i)9-s − 0.184i·10-s + 0.801·11-s + (−0.763 − 0.518i)12-s − 0.340·13-s + 0.261·14-s + (−0.550 − 0.373i)15-s + 0.774·16-s + 1.37·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$69$$    =    $$3 \cdot 23$$ Sign: $0.801 + 0.598i$ Analytic conductor: $$11.0664$$ Root analytic conductor: $$3.32663$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{69} (68, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 69,\ (\ :5/2),\ 0.801 + 0.598i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.87217 - 0.621919i$$ $$L(\frac12)$$ $$\approx$$ $$1.87217 - 0.621919i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (12.8 + 8.75i)T$$
23 $$1 + (-2.53e3 - 114. i)T$$
good2 $$1 + 1.57iT - 32T^{2}$$
5 $$1 - 37.1T + 3.12e3T^{2}$$
7 $$1 - 122. iT - 1.68e4T^{2}$$
11 $$1 - 321.T + 1.61e5T^{2}$$
13 $$1 + 207.T + 3.71e5T^{2}$$
17 $$1 - 1.64e3T + 1.41e6T^{2}$$
19 $$1 + 2.88e3iT - 2.47e6T^{2}$$
29 $$1 - 2.44e3iT - 2.05e7T^{2}$$
31 $$1 + 244.T + 2.86e7T^{2}$$
37 $$1 - 9.93e3iT - 6.93e7T^{2}$$
41 $$1 + 1.74e4iT - 1.15e8T^{2}$$
43 $$1 - 3.35e3iT - 1.47e8T^{2}$$
47 $$1 - 1.50e4iT - 2.29e8T^{2}$$
53 $$1 + 7.91e3T + 4.18e8T^{2}$$
59 $$1 - 1.13e4iT - 7.14e8T^{2}$$
61 $$1 - 2.59e4iT - 8.44e8T^{2}$$
67 $$1 - 3.69e4iT - 1.35e9T^{2}$$
71 $$1 + 5.28e4iT - 1.80e9T^{2}$$
73 $$1 + 7.18e4T + 2.07e9T^{2}$$
79 $$1 + 4.33e4iT - 3.07e9T^{2}$$
83 $$1 - 1.07e5T + 3.93e9T^{2}$$
89 $$1 + 1.12e3T + 5.58e9T^{2}$$
97 $$1 - 2.67e4iT - 8.58e9T^{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

−13.34355681568220850291656190288, −12.22577182953355286199127186755, −11.63556376222812134578164724703, −10.50460080084017646535399169015, −9.209748798088403698032457988499, −7.36067775554720037958740733370, −6.33079170569334315500524575570, −5.28702331187473747630531701625, −2.66595880611685887169020878511, −1.27893399997193357876330113620, 1.33206432112357111356092214367, 3.69915357373700703979230745493, 5.50407786939411325281237690539, 6.45653102874011109615547367421, 7.66994981520401064824369656423, 9.723064361413992759195426562312, 10.41881503315544351542197839863, 11.55821821948312214714163235709, 12.49804271744781678651409841424, 14.18509862942352499187353455842