# Properties

 Label 2-40e2-20.3-c1-0-14 Degree $2$ Conductor $1600$ Sign $0.850 - 0.525i$ Analytic cond. $12.7760$ Root an. cond. $3.57436$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## 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 & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1600$$    =    $$2^{6} \cdot 5^{2}$$ Sign: $0.850 - 0.525i$ Analytic conductor: $$12.7760$$ Root analytic conductor: $$3.57436$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1600} (1343, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1600,\ (\ :1/2),\ 0.850 - 0.525i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.356187041$$ $$L(\frac12)$$ $$\approx$$ $$2.356187041$$ $$L(\frac{3}{2})$$ 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

−8.949308776173010957975980837711, −8.615438685139658252791054915969, −8.217952852428854810315473712050, −7.11841544533514631319757746441, −6.31477005881578345439531718352, −5.35193605369363449701521096422, −4.57260612362092408740704908740, −3.30927135895031105960132761393, −2.17910011568493508616522677277, −1.56039085313696789733358049108, 0.911043372697545569805261962514, 2.24730671604403008144562798173, 3.58089537743508553096294215741, 4.18423420983815244687014448918, 4.94092230917738424743200509192, 6.11097635043789657226929159402, 7.07947089281660569271320211210, 7.925309679445865231984154554998, 8.454380564982761455140661602334, 9.363258489822003334166133824785