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Properties

Label 1-123-123.29-r0-0-0
Degree $1$
Conductor $123$
Sign $0.646 + 0.762i$
Analytic cond. $0.571209$
Root an. cond. $0.571209$
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (0.453 − 0.891i)7-s + (0.587 + 0.809i)8-s + (−0.809 + 0.587i)10-s + (0.987 + 0.156i)11-s + (−0.891 + 0.453i)13-s + (0.707 − 0.707i)14-s + (0.309 + 0.951i)16-s + (−0.156 + 0.987i)17-s + (−0.891 − 0.453i)19-s + (−0.951 + 0.309i)20-s + (0.891 + 0.453i)22-s + (0.309 − 0.951i)23-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (0.453 − 0.891i)7-s + (0.587 + 0.809i)8-s + (−0.809 + 0.587i)10-s + (0.987 + 0.156i)11-s + (−0.891 + 0.453i)13-s + (0.707 − 0.707i)14-s + (0.309 + 0.951i)16-s + (−0.156 + 0.987i)17-s + (−0.891 − 0.453i)19-s + (−0.951 + 0.309i)20-s + (0.891 + 0.453i)22-s + (0.309 − 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(123\)    =    \(3 \cdot 41\)
Sign: $0.646 + 0.762i$
Analytic conductor: \(0.571209\)
Root analytic conductor: \(0.571209\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{123} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 123,\ (0:\ ),\ 0.646 + 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.579617716 + 0.7313896605i\)
\(L(\frac12)\) \(\approx\) \(1.579617716 + 0.7313896605i\)
\(L(1)\) \(\approx\) \(1.578423360 + 0.4780563989i\)
\(L(1)\) \(\approx\) \(1.578423360 + 0.4780563989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (0.453 - 0.891i)T \)
11 \( 1 + (0.987 + 0.156i)T \)
13 \( 1 + (-0.891 + 0.453i)T \)
17 \( 1 + (-0.156 + 0.987i)T \)
19 \( 1 + (-0.891 - 0.453i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.156 - 0.987i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (-0.453 - 0.891i)T \)
53 \( 1 + (0.156 + 0.987i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.951 - 0.309i)T \)
67 \( 1 + (0.987 - 0.156i)T \)
71 \( 1 + (-0.987 - 0.156i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.453 + 0.891i)T \)
97 \( 1 + (-0.987 + 0.156i)T \)
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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

−29.020221667691053177720987101113, −27.767275839391161317500098077493, −27.36495976756392993944810389953, −25.19079075113120349599439711687, −24.74058331787966581176655273585, −23.79799061770691114918661363497, −22.69403268742841957978129183951, −21.75935582494264678821640423643, −20.804948355105498842032329281975, −19.80446250848178637621378780383, −19.01566549697204145110518281620, −17.345697453333652336465789907199, −16.12808909039296462738311587293, −15.17234414035532968685580411812, −14.28735600448484856632575357329, −12.87004219587056638484508491172, −12.02531931062200257186013746479, −11.350694282176516203142821289501, −9.662069511883524874103724050050, −8.397070183692170336112630167012, −6.932409604211364244878950173488, −5.41739656042121209056743047050, −4.61508062564556267715743186890, −3.18538188829934905261007857605, −1.58485135409275560578355174619, 2.216217106214557756546448278349, 3.83890188343656785610818117368, 4.527358027068203680821039671990, 6.43176062134233362904004472560, 7.12415037680570564070281589933, 8.308017877759725026288763799717, 10.34534447815966248241539624064, 11.32760073071612341742165243422, 12.26951239729487308696827134561, 13.65377665081242618272357344746, 14.653009545470097121291296722231, 15.164455406962130918049779264905, 16.75857056428981369483966808721, 17.37580527080695524076285150602, 19.221631422505042552173254501944, 19.94931705795145652384064407497, 21.23703091979372101971549274644, 22.22160598082769627774577917609, 23.05091077159668501374203733520, 23.932415937489970546705780675459, 24.80562355832116433345978190868, 26.2428369491747782086020704718, 26.72258531094258188894574330515, 28.09885303880665814334489912467, 29.695297699686929905141280427802

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