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Properties

Label 1-23-23.6-r0-0-0
Degree $1$
Conductor $23$
Sign $0.854 - 0.519i$
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  + (−0.654 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(23\)
Sign: $0.854 - 0.519i$
Analytic conductor: \(0.106811\)
Root analytic conductor: \(0.106811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 23,\ (0:\ ),\ 0.854 - 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5936953919 - 0.1662737264i\)
\(L(\frac12)\) \(\approx\) \(0.5936953919 - 0.1662737264i\)
\(L(1)\) \(\approx\) \(0.8044174782 - 0.1800159172i\)
\(L(1)\) \(\approx\) \(0.8044174782 - 0.1800159172i\)

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

−38.700408365765894581394546516791, −37.374582052745730403293364755146, −36.73684444152570151931867284665, −35.19931983131626181222038711038, −34.39659421649262485665831178230, −32.686616841364382480753785584274, −31.68697678664476224776710908677, −29.90051658251202674860051032003, −28.79736871063084622912026153343, −26.68424173530906038326597384210, −26.03410951890411644829689228165, −24.959998697416946641111554588972, −23.61823174774936248044005691048, −21.899117064346872748693005649887, −19.63149511760766360944426352873, −18.86079323503127356346653185205, −17.60123837809627875346084807219, −15.65730440476089251256326484275, −14.4875947273727442066818885853, −13.12631838826265572682657003615, −10.40381591166334108468080526150, −9.0236073970210270980590936589, −7.386736091739146298255844220707, −6.13271706638252198914368019481, −2.695063685459656135537545911260, 2.47039178966832528664961712539, 4.432117621973441540755241160510, 7.720799325920089958960128054403, 9.39325673406057682914387718310, 10.03959677418238791346100433673, 12.425109771331570931137449711022, 13.61944875373012828405910231195, 15.89863210493409896148305765933, 17.07638010143655944598506026097, 18.98899807308690491928557869720, 20.21715299352348244292115288302, 20.977937859874896897254665650164, 22.4678459530138552535662630855, 24.93649520853625830521165936466, 25.9574329561884069176049822057, 27.13223980683776795140294166042, 28.50879775493850333463910445784, 29.53655072465957060785288875489, 31.32056718767089627329744047203, 32.1091735901850118059187416565, 33.703928987552825404236293326258, 35.938907075447944483317436388978, 36.27341873101567924360620203011, 37.635050660222910611763365632394, 38.82904969900439216797491855283

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