# Properties

 Label 2-800-20.7-c1-0-15 Degree $2$ Conductor $800$ Sign $-0.525 + 0.850i$ Analytic cond. $6.38803$ Root an. cond. $2.52745$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 − i)3-s + (3 − 3i)7-s − i·9-s − 2i·11-s + (−3 + 3i)13-s + (−1 − i)17-s + 4·19-s − 6·21-s + (1 + i)23-s + (−4 + 4i)27-s − 10i·31-s + (−2 + 2i)33-s + (1 + i)37-s + 6·39-s − 10·41-s + ⋯
 L(s)  = 1 + (−0.577 − 0.577i)3-s + (1.13 − 1.13i)7-s − 0.333i·9-s − 0.603i·11-s + (−0.832 + 0.832i)13-s + (−0.242 − 0.242i)17-s + 0.917·19-s − 1.30·21-s + (0.208 + 0.208i)23-s + (−0.769 + 0.769i)27-s − 1.79i·31-s + (−0.348 + 0.348i)33-s + (0.164 + 0.164i)37-s + 0.960·39-s − 1.56·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$800$$    =    $$2^{5} \cdot 5^{2}$$ Sign: $-0.525 + 0.850i$ Analytic conductor: $$6.38803$$ Root analytic conductor: $$2.52745$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{800} (607, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 800,\ (\ :1/2),\ -0.525 + 0.850i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.573689 - 1.02897i$$ $$L(\frac12)$$ $$\approx$$ $$0.573689 - 1.02897i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + (1 + i)T + 3iT^{2}$$
7 $$1 + (-3 + 3i)T - 7iT^{2}$$
11 $$1 + 2iT - 11T^{2}$$
13 $$1 + (3 - 3i)T - 13iT^{2}$$
17 $$1 + (1 + i)T + 17iT^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 + (-1 - i)T + 23iT^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 10iT - 31T^{2}$$
37 $$1 + (-1 - i)T + 37iT^{2}$$
41 $$1 + 10T + 41T^{2}$$
43 $$1 + (5 + 5i)T + 43iT^{2}$$
47 $$1 + (-3 + 3i)T - 47iT^{2}$$
53 $$1 + (-5 + 5i)T - 53iT^{2}$$
59 $$1 + 12T + 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 + (-1 + i)T - 67iT^{2}$$
71 $$1 + 2iT - 71T^{2}$$
73 $$1 + (1 - i)T - 73iT^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 + (5 + 5i)T + 83iT^{2}$$
89 $$1 - 16iT - 89T^{2}$$
97 $$1 + (-3 - 3i)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

−10.04601612046077669039541069461, −9.177523187138729099815204687783, −8.043369381561794008056577988471, −7.29577776853698128231862495224, −6.68522468942190639094060403261, −5.51986834619945442717582828569, −4.63355241801052051179115717180, −3.58560007532189503648850603494, −1.86629275310895654054021106238, −0.64506924347125637965437535748, 1.78704947084357478927538519404, 2.98583320373619828613211231407, 4.70724086947760161543118487461, 5.04279350927061140336413948223, 5.84321312207294119439341714655, 7.20776009761738679295576243310, 8.055300510133531601149933734628, 8.845966807037717208827539248878, 9.881633541073524618857510399796, 10.53801898130760231602550792017