# Properties

 Label 2-819-117.41-c1-0-42 Degree $2$ Conductor $819$ Sign $7.70e-5 - 0.999i$ Analytic cond. $6.53974$ Root an. cond. $2.55729$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.06 + 0.285i)2-s + (0.611 + 1.62i)3-s + (−0.677 + 0.391i)4-s + (4.22 − 1.13i)5-s + (−1.11 − 1.55i)6-s + (0.707 + 0.707i)7-s + (2.17 − 2.17i)8-s + (−2.25 + 1.98i)9-s + (−4.18 + 2.41i)10-s + (−0.432 − 1.61i)11-s + (−1.04 − 0.859i)12-s + (−2.06 + 2.95i)13-s + (−0.955 − 0.551i)14-s + (4.41 + 6.15i)15-s + (−0.910 + 1.57i)16-s + (−0.720 + 1.24i)17-s + ⋯
 L(s)  = 1 + (−0.753 + 0.201i)2-s + (0.352 + 0.935i)3-s + (−0.338 + 0.195i)4-s + (1.89 − 0.506i)5-s + (−0.454 − 0.633i)6-s + (0.267 + 0.267i)7-s + (0.767 − 0.767i)8-s + (−0.750 + 0.660i)9-s + (−1.32 + 0.763i)10-s + (−0.130 − 0.486i)11-s + (−0.302 − 0.248i)12-s + (−0.571 + 0.820i)13-s + (−0.255 − 0.147i)14-s + (1.14 + 1.59i)15-s + (−0.227 + 0.394i)16-s + (−0.174 + 0.302i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (7.70e-5 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (7.70e-5 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$819$$    =    $$3^{2} \cdot 7 \cdot 13$$ Sign: $7.70e-5 - 0.999i$ Analytic conductor: $$6.53974$$ Root analytic conductor: $$2.55729$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{819} (743, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 819,\ (\ :1/2),\ 7.70e-5 - 0.999i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.02022 + 1.02015i$$ $$L(\frac12)$$ $$\approx$$ $$1.02022 + 1.02015i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.611 - 1.62i)T$$
7 $$1 + (-0.707 - 0.707i)T$$
13 $$1 + (2.06 - 2.95i)T$$
good2 $$1 + (1.06 - 0.285i)T + (1.73 - i)T^{2}$$
5 $$1 + (-4.22 + 1.13i)T + (4.33 - 2.5i)T^{2}$$
11 $$1 + (0.432 + 1.61i)T + (-9.52 + 5.5i)T^{2}$$
17 $$1 + (0.720 - 1.24i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-1.10 - 4.11i)T + (-16.4 + 9.5i)T^{2}$$
23 $$1 - 7.17T + 23T^{2}$$
29 $$1 + (0.483 + 0.278i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (0.393 + 1.46i)T + (-26.8 + 15.5i)T^{2}$$
37 $$1 + (1.95 - 7.30i)T + (-32.0 - 18.5i)T^{2}$$
41 $$1 + (2.28 + 2.28i)T + 41iT^{2}$$
43 $$1 + 6.80iT - 43T^{2}$$
47 $$1 + (-9.28 - 2.48i)T + (40.7 + 23.5i)T^{2}$$
53 $$1 - 13.5iT - 53T^{2}$$
59 $$1 + (-2.71 - 0.727i)T + (51.0 + 29.5i)T^{2}$$
61 $$1 - 8.52T + 61T^{2}$$
67 $$1 + (1.15 - 1.15i)T - 67iT^{2}$$
71 $$1 + (13.2 - 3.55i)T + (61.4 - 35.5i)T^{2}$$
73 $$1 + (5.23 + 5.23i)T + 73iT^{2}$$
79 $$1 + (2.40 + 4.16i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-1.06 + 3.96i)T + (-71.8 - 41.5i)T^{2}$$
89 $$1 + (4.58 + 1.22i)T + (77.0 + 44.5i)T^{2}$$
97 $$1 + (8.60 - 8.60i)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

−10.18373401108473510344374722609, −9.409092208365469702856551401061, −8.954305811057150871626755958182, −8.422790792274684525027753584007, −7.13977463350010468919087186226, −5.88579933896112867730311649719, −5.14858397632245007210374713529, −4.27696626870025237262474949085, −2.78558784404552648141355328769, −1.52184989034376871586110226885, 0.989085260514085523731756557831, 2.08683707146917313169486743557, 2.85405048803790366004294752530, 5.01777716410934072243560385890, 5.59152143425681664052025025818, 6.85669009111746061361570103788, 7.34902479372710785230805884341, 8.579019890242253502877323045787, 9.253742558098165048175979021259, 9.860271987721298202553892340171