Properties

Degree 1
Conductor 23
Sign $0.0117 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (−0.654 + 0.755i)5-s + (−0.959 − 0.281i)6-s + (0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (−0.142 + 0.989i)11-s + (−0.142 + 0.989i)12-s + (0.841 + 0.540i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + ⋯
L(s,χ)  = 1  + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (−0.654 + 0.755i)5-s + (−0.959 − 0.281i)6-s + (0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (−0.142 + 0.989i)11-s + (−0.142 + 0.989i)12-s + (0.841 + 0.540i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0117 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0117 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $0.0117 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{23} (16, \cdot )$
Sato-Tate  :  $\mu(11)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 23,\ (0:\ ),\ 0.0117 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4542820890 - 0.4596636635i$
$L(\frac12,\chi)$  $\approx$  $0.4542820890 - 0.4596636635i$
$L(\chi,1)$  $\approx$  0.7124830147 - 0.4910059593i
$L(1,\chi)$  $\approx$  0.7124830147 - 0.4910059593i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−39.630589253427970962995821234711, −37.77411630115861677116390700405, −36.83136652728095244607274574675, −35.31986669431009589084467665916, −34.187775530292962937658311082520, −32.87424424000170917884878324688, −31.87412179805184831834560608843, −30.97805367598638152476017439155, −28.1210607618274055065119894514, −27.50777201752062610708812593831, −26.275631990238364246164021111486, −24.83033504983248348613401784470, −23.76188687440743738536230743016, −22.066231640167673219479742483795, −20.67494430032699797550929546084, −19.00505634356831335793236361495, −17.16891944881079233650210363882, −15.85253243312657114363761589162, −15.04251051590667695399815539530, −13.380213450669045687542426834143, −11.00692064811433854564371405694, −8.828560229821020532492420431612, −8.23455744973127504023626840065, −5.527258524941725784466998943930, −4.12168134698006821351736110537, 2.044762235917227441005602451891, 4.03528117503497395606199794164, 7.178383719762419099693442656676, 8.580891341384805742436590726156, 10.75645338483036295494981316768, 11.94789351008338603970116213884, 13.52059537639149946355024865231, 14.77864637462672441075329062038, 17.5871153342747464320750956621, 18.55761841830637896742463096719, 19.77526761173815985849828230239, 20.87208035387172726679522644384, 22.84970431975498903297629724134, 23.75888055708372168717217522032, 25.83869469482731532443275203768, 26.96277899701958235711638522869, 28.42778028807417555980305445107, 30.00548137405077635256440342745, 30.65281967237935792977330619243, 31.51461797195130987311578820832, 33.71450450824251765414541776951, 35.405010925966572274206402222, 36.23026982523501020290933788759, 37.5173950391724365292793859954, 38.50340307558800926550590898790

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