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Properties

Label 1-23-23.8-r0-0-0
Degree $1$
Conductor $23$
Sign $0.381 + 0.924i$
Analytic cond. $0.106811$
Root an. cond. $0.106811$
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  + (−0.959 − 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.841 − 0.540i)6-s + (0.415 + 0.909i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−0.959 + 0.281i)11-s + (−0.959 + 0.281i)12-s + (0.415 − 0.909i)13-s + (−0.142 − 0.989i)14-s + (−0.654 − 0.755i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.841 − 0.540i)6-s + (0.415 + 0.909i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−0.959 + 0.281i)11-s + (−0.959 + 0.281i)12-s + (0.415 − 0.909i)13-s + (−0.142 − 0.989i)14-s + (−0.654 − 0.755i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(23\)
Sign: $0.381 + 0.924i$
Analytic conductor: \(0.106811\)
Root analytic conductor: \(0.106811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 23,\ (0:\ ),\ 0.381 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3277273320 + 0.2192356379i\)
\(L(\frac12)\) \(\approx\) \(0.3277273320 + 0.2192356379i\)
\(L(1)\) \(\approx\) \(0.5201889752 + 0.1855913550i\)
\(L(1)\) \(\approx\) \(0.5201889752 + 0.1855913550i\)

Euler product

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

−38.98998143260861314826998406865, −36.76342345208191279745533848015, −36.345291181608239217906100369, −35.13830079197997026787697269749, −33.97287555764827752484027603333, −32.78621332865303291066825772579, −30.723716614579539498499726230653, −29.195470169184944176971213074104, −28.45233492006461977443780344521, −27.14720500155539862608126843923, −25.58753454839631486009922192740, −23.90697140200119864816739964174, −23.746162420970419735183934618656, −20.98126750068120979775972839618, −19.62895355377679179633632731038, −18.24509377294006263087636494940, −16.99597997769593897315678826565, −16.125151922336169579796691386233, −13.69354664177652326072264669422, −11.92528411857715341626447780780, −10.52002687418432590322427583875, −8.43825346336987519118210847845, −7.19217904241057344185175816264, −5.31655872189447821089738407592, −1.27653878245145980082795619280, 3.01197442570287685806567374878, 5.78010856069087742744357667992, 7.81320610194552651361795590287, 9.768718420132840337553343315027, 10.88483862655997164205213863176, 12.0787049912967822951052893076, 15.06133768301543924413000424979, 16.02772464174142797019205408389, 17.87536071873842841425803508647, 18.52349463683331496984900599528, 20.58110386643468482917126950490, 21.69860673246791804470920090761, 23.07320963138608532764277937631, 25.19677779353309866265573337361, 26.497839210692829050247391031, 27.526893336742595010903382585128, 28.54104978554951604586887868863, 29.85958297183451124905676931097, 31.32850806200926789795211473721, 33.43445248621830591262689882652, 34.34335952807736822061983349341, 35.09279704228361515166147371397, 37.07764318936724965380662137499, 38.04978520853081323510024006482, 38.86901206720230220250172177424

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