# Properties

 Label 2-1352-104.3-c0-0-7 Degree $2$ Conductor $1352$ Sign $0.488 + 0.872i$ Analytic cond. $0.674735$ Root an. cond. $0.821423$ Motivic weight $0$ 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 + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − i·5-s + (0.866 + 0.499i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.5 − 0.866i)10-s + 0.999·12-s − 0.999·14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.499i)20-s − 0.999i·21-s + ⋯
 L(s)  = 1 + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − i·5-s + (0.866 + 0.499i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.5 − 0.866i)10-s + 0.999·12-s − 0.999·14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.499i)20-s − 0.999i·21-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1352$$    =    $$2^{3} \cdot 13^{2}$$ Sign: $0.488 + 0.872i$ Analytic conductor: $$0.674735$$ Root analytic conductor: $$0.821423$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1352} (315, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1352,\ (\ :0),\ 0.488 + 0.872i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.902426545$$ $$L(\frac12)$$ $$\approx$$ $$1.902426545$$ $$L(1)$$ 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$$
13 $$1$$
good3 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
5 $$1 + iT - T^{2}$$
7 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
11 $$1 + (-0.5 + 0.866i)T^{2}$$
17 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
19 $$1 + (-0.5 - 0.866i)T^{2}$$
23 $$1 + (0.5 - 0.866i)T^{2}$$
29 $$1 + (0.5 - 0.866i)T^{2}$$
31 $$1 - 2iT - T^{2}$$
37 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
41 $$1 + (-0.5 + 0.866i)T^{2}$$
43 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
47 $$1 - iT - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 + (-0.5 - 0.866i)T^{2}$$
61 $$1 + (0.5 + 0.866i)T^{2}$$
67 $$1 + (-0.5 + 0.866i)T^{2}$$
71 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
73 $$1 + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + (-0.5 + 0.866i)T^{2}$$
97 $$1 + (-0.5 - 0.866i)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

−9.718282037282881841020902756979, −9.140190807241498300615741095123, −8.321905772282808814128531332363, −6.94955430246636260664819329260, −6.23423146601742820600109286327, −5.10879997575155783910310188429, −4.39561646553212816775993118578, −3.71196760942098020342955847462, −2.87827314733881036786411533067, −1.29676769859631177276471469427, 2.29428874240850158571732940594, 2.75626672508052850615022171515, 3.75559546066981199487332866595, 4.98053795137731331921032491752, 6.10944561910638960081987133091, 6.65520050365649572652023475850, 7.30157786417098438273091637500, 7.971085016617006416533514411200, 8.924852405632046828378905518650, 9.878223563661038460325055686557