Properties

Degree 1
Conductor 23
Sign $0.102 + 0.994i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.102 + 0.994i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.102 + 0.994i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $0.102 + 0.994i$
motivic weight  =  \(0\)
character  :  $\chi_{23} (2, \cdot )$
Sato-Tate  :  $\mu(11)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 23,\ (0:\ ),\ 0.102 + 0.994i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4984334565 + 0.4498776534i$
$L(\frac12,\chi)$  $\approx$  $0.4984334565 + 0.4498776534i$
$L(\chi,1)$  $\approx$  0.7656286297 + 0.4719765327i
$L(1,\chi)$  $\approx$  0.7656286297 + 0.4719765327i

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

−38.93227603425784446495952689359, −37.64344132096444354230784625299, −36.34034860653278949619030657922, −35.01123939435314314699907878999, −33.32066917719552179756545379953, −32.323690816776620966386330634248, −30.66562812414742044523994530864, −29.498898799058772541138821747462, −28.38890871123083427352973893709, −27.79152477561257132364759370303, −25.20905107559983377504372910324, −23.907956068594960815477820614104, −22.34674159030419039395523676353, −21.651087899353085451831570335123, −20.00084804266995032587200308355, −18.2953184608830274021420282350, −17.33455318923654554998944279741, −15.15861404994762044510254589809, −13.03008807756220258135985282913, −12.35530432162033183302109670094, −10.67742555619709718993397773164, −9.19792677852480228595721826582, −6.094388119466841465630337701608, −4.880221735492892570773084241790, −1.96977154104919041139019471733, 4.107576373454723068567157229695, 5.93629800683368907367826622427, 6.99445687720755055377747596611, 9.518473151719553072761703782662, 11.2404244575341070408591752866, 13.25299999171408835935237833687, 14.46819153630720112687534977385, 16.38235179737803385760600853773, 17.12043426553161151564416818047, 18.49347494975654967418709777481, 21.22473102294776962351969816040, 22.23500248563268067911829484139, 23.335879007361754699588913264540, 24.59672044392344343168085403650, 26.29265029361162290990550266473, 27.117727950932537400599384314731, 29.16979734371197588996575386782, 30.09211064571329568365542947311, 32.05532417944817604224827698349, 33.37162233279326985283445460832, 33.75938598263804897594035083041, 35.19128686971259801698748864313, 36.43158391733212312676607291266, 38.30686790930825263743043829346, 39.83187983449892004282309558908

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