Properties

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

Related objects

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Normalization:  

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

Graph of the $Z$-function along the critical line