# Properties

 Degree $1$ Conductor $43$ Sign $1$ Motivic weight $0$ Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(χ,s)  = 1 − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯
 L(s,χ)  = 1 − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(\chi,s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}
\begin{aligned}\Lambda(s,\chi)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$43$$ Sign: $1$ Motivic weight: $$0$$ Character: $\chi_{43} (42, \cdot )$ Sato-Tate group: $\mu(2)$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 43,\ (1:\ ),\ 1)$$

## Particular Values

 $$L(\chi,\frac{1}{2})$$ $$\approx$$ $$0.5247737459$$ $$L(\frac12,\chi)$$ $$\approx$$ $$0.5247737459$$ $$L(\chi,1)$$ $$\approx$$ $$0.4790883882$$ $$L(1,\chi)$$ $$\approx$$ $$0.4790883882$$

## Euler product

$$L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}$$
$$L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−34.83092840169961121719144213552, −33.4927797237544288292450796231, −32.3101068295994425756839252687, −30.357862018896409935427709001215, −29.464776898064711140571970778951, −28.09335583171493624603559604114, −27.65749009246467670308015460132, −26.323411789686364133656778792272, −24.98922145195416755611140185811, −23.55878453645455334291315462732, −22.64296312368434067990202224085, −20.96389199232680804233909915734, −19.358878795253897913047146670926, −18.74862050556175441281264360145, −17.0873796669794460979342884775, −16.30373940188503475195525493041, −15.24950229092514405157738797122, −12.606143870467470331509577341313, −11.56601380720052741703438312429, −10.44597500315359762114740100317, −8.92663404729950315681373419887, −7.20442176171099355167338900362, −6.1302234797929222268686196454, −3.695326872142827304010572169930, −0.836400774360507167568100327000, 0.836400774360507167568100327000, 3.695326872142827304010572169930, 6.1302234797929222268686196454, 7.20442176171099355167338900362, 8.92663404729950315681373419887, 10.44597500315359762114740100317, 11.56601380720052741703438312429, 12.606143870467470331509577341313, 15.24950229092514405157738797122, 16.30373940188503475195525493041, 17.0873796669794460979342884775, 18.74862050556175441281264360145, 19.358878795253897913047146670926, 20.96389199232680804233909915734, 22.64296312368434067990202224085, 23.55878453645455334291315462732, 24.98922145195416755611140185811, 26.323411789686364133656778792272, 27.65749009246467670308015460132, 28.09335583171493624603559604114, 29.464776898064711140571970778951, 30.357862018896409935427709001215, 32.3101068295994425756839252687, 33.4927797237544288292450796231, 34.83092840169961121719144213552