# Properties

 Label 2-4600-5.4-c1-0-40 Degree $2$ Conductor $4600$ Sign $0.447 - 0.894i$ Analytic cond. $36.7311$ Root an. cond. $6.06062$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s + 2·9-s + 2·11-s + 5i·13-s − 4i·17-s + 2·19-s + i·23-s + 5i·27-s + 3·29-s + 7·31-s + 2i·33-s − 2i·37-s − 5·39-s − 9·41-s + 4i·43-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 0.666·9-s + 0.603·11-s + 1.38i·13-s − 0.970i·17-s + 0.458·19-s + 0.208i·23-s + 0.962i·27-s + 0.557·29-s + 1.25·31-s + 0.348i·33-s − 0.328i·37-s − 0.800·39-s − 1.40·41-s + 0.609i·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4600$$    =    $$2^{3} \cdot 5^{2} \cdot 23$$ Sign: $0.447 - 0.894i$ Analytic conductor: $$36.7311$$ Root analytic conductor: $$6.06062$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{4600} (4049, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4600,\ (\ :1/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.239291412$$ $$L(\frac12)$$ $$\approx$$ $$2.239291412$$ $$L(\frac{3}{2})$$ 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$$
23 $$1 - iT$$
good3 $$1 - iT - 3T^{2}$$
7 $$1 - 7T^{2}$$
11 $$1 - 2T + 11T^{2}$$
13 $$1 - 5iT - 13T^{2}$$
17 $$1 + 4iT - 17T^{2}$$
19 $$1 - 2T + 19T^{2}$$
29 $$1 - 3T + 29T^{2}$$
31 $$1 - 7T + 31T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 + 9T + 41T^{2}$$
43 $$1 - 4iT - 43T^{2}$$
47 $$1 + 9iT - 47T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 + 2iT - 67T^{2}$$
71 $$1 + T + 71T^{2}$$
73 $$1 + iT - 73T^{2}$$
79 $$1 - 14T + 79T^{2}$$
83 $$1 - 83T^{2}$$
89 $$1 + 16T + 89T^{2}$$
97 $$1 + 4iT - 97T^{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

−8.659274547511491376797276220149, −7.63038858744467132932930325995, −6.90025614703684659844922500466, −6.47325099482051362561726648766, −5.32107833308116842769210031163, −4.62941543333005518116565524367, −4.05583855104512051400844996245, −3.20122168115195979571617579194, −2.07597613751231664573489242707, −1.04584270128031457834865558683, 0.76274912844162435656476452823, 1.59101803780633745754875348953, 2.70129889536784819637221992176, 3.59259555539317274659978741662, 4.43446087846467611734654205892, 5.30802100754912077238355864120, 6.16947130669112473693139586873, 6.71432553154183620787476990508, 7.49242049364059786024122014112, 8.179601580147223344006452475914