# Properties

 Label 2-1638-13.10-c1-0-25 Degree $2$ Conductor $1638$ Sign $0.287 + 0.957i$ Analytic cond. $13.0794$ Root an. cond. $3.61655$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − 1.25i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (0.627 + 1.08i)10-s + (4.35 − 2.51i)11-s + (−3.43 + 1.08i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (2.26 − 3.91i)17-s + (−3.05 − 1.76i)19-s + (−1.08 − 0.627i)20-s + (−2.51 + 4.35i)22-s + (−2.56 − 4.44i)23-s + ⋯
 L(s)  = 1 + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.561i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (0.198 + 0.343i)10-s + (1.31 − 0.757i)11-s + (−0.953 + 0.299i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.548 − 0.949i)17-s + (−0.700 − 0.404i)19-s + (−0.243 − 0.140i)20-s + (−0.535 + 0.928i)22-s + (−0.535 − 0.927i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1638$$    =    $$2 \cdot 3^{2} \cdot 7 \cdot 13$$ Sign: $0.287 + 0.957i$ Analytic conductor: $$13.0794$$ Root analytic conductor: $$3.61655$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1638} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1638,\ (\ :1/2),\ 0.287 + 0.957i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.124423492$$ $$L(\frac12)$$ $$\approx$$ $$1.124423492$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.866 - 0.5i)T$$
3 $$1$$
7 $$1 + (-0.866 - 0.5i)T$$
13 $$1 + (3.43 - 1.08i)T$$
good5 $$1 + 1.25iT - 5T^{2}$$
11 $$1 + (-4.35 + 2.51i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 + (-2.26 + 3.91i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (3.05 + 1.76i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (2.56 + 4.44i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-0.765 - 1.32i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + 0.439iT - 31T^{2}$$
37 $$1 + (8.90 - 5.13i)T + (18.5 - 32.0i)T^{2}$$
41 $$1 + (-8.65 + 4.99i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (1.83 - 3.17i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 - 3.60iT - 47T^{2}$$
53 $$1 + 8.82T + 53T^{2}$$
59 $$1 + (-0.542 - 0.313i)T + (29.5 + 51.0i)T^{2}$$
61 $$1 + (-4.49 + 7.79i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-6.40 + 3.69i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (5.89 + 3.40i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 + 11.5iT - 73T^{2}$$
79 $$1 + 6.74T + 79T^{2}$$
83 $$1 + 9.57iT - 83T^{2}$$
89 $$1 + (-7.28 + 4.20i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 + (2.43 + 1.40i)T + (48.5 + 84.0i)T^{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

−9.042303568097802109442459921568, −8.624307915318552219475469459179, −7.71939991882906642577516743846, −6.82404768351502607700422487191, −6.17809075574567712304197797033, −5.07913960842831024193309362748, −4.43998190289900610182154746102, −3.05834199374618760865495683949, −1.78751009987588553996505260975, −0.56020374019418857421822118974, 1.37566802431133529661062055954, 2.31384888307251556914184640873, 3.57724943123462653509897394998, 4.28305775953619292799277699795, 5.53979410659511598123027094602, 6.59678534884718205820060887593, 7.21642504759040086178011827870, 7.977081832223158844724820017107, 8.815250495253581740176245987184, 9.699613058796789436255358358449