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Properties

Label 1-23-23.19-r1-0-0
Degree $1$
Conductor $23$
Sign $0.915 - 0.403i$
Analytic cond. $2.47169$
Root an. cond. $2.47169$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(23\)
Sign: $0.915 - 0.403i$
Analytic conductor: \(2.47169\)
Root analytic conductor: \(2.47169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 23,\ (1:\ ),\ 0.915 - 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193390215 - 0.2513170686i\)
\(L(\frac12)\) \(\approx\) \(1.193390215 - 0.2513170686i\)
\(L(1)\) \(\approx\) \(1.024776888 - 0.06836387700i\)
\(L(1)\) \(\approx\) \(1.024776888 - 0.06836387700i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.654 + 0.755i)T \)
3 \( 1 + (0.841 - 0.540i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (-0.959 - 0.281i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (-0.142 + 0.989i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (-0.415 + 0.909i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (-0.415 + 0.909i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−38.52346429453796222459047680066, −37.46216404834207596121828809078, −36.85009267046127835910036362665, −35.04956605322416709788179171930, −33.93016426705979391415339165970, −31.97348917347789808407302969171, −30.74581808560397652685458492907, −30.01430159849003592791241125303, −27.988419931107052543334527769683, −26.951390997049195127006939735059, −26.21775611958856808744246734444, −24.55984974550267442921257070393, −22.11537274776449995636983231261, −21.3261356712269760072526249366, −19.71961408220323290488564258838, −18.89371584858930917798322621504, −17.20644616703920879527303097668, −15.28336328858085073315065112200, −13.96328593621444567575480909438, −11.70549832767387652110164846562, −10.489122351861352389188803100170, −8.8795589013945002698155122045, −7.57658983559309863798267647923, −4.06368017958575994735722742671, −2.422807249193995525545659390855, 1.3797846159960658038612160980, 4.76819409837205579260399461408, 7.24691052521206073122623024527, 8.26915988698542440107253097219, 9.63430524416145883796502627340, 12.11256698180227244551608080116, 14.04445956117997645153264879685, 15.140311256924691708540367847441, 16.874925313993120683214553437949, 18.14111882842997627545344585960, 19.725151936370181535441078356765, 20.539451934818267454102740931592, 23.274507660989997587087547744148, 24.53571898560322130309389782054, 25.085663704080201694499223054965, 26.86628733966501129913588702712, 27.70102760915420388331662960571, 29.44533162742435653149063788306, 31.143738407768407904012382129503, 32.202160011262934899636313295659, 33.560340825986917411840731135035, 35.13458641489215004260488820128, 36.15326235998923348993679305748, 36.85880858660055521816147706964, 38.199345952938279834194250800475

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