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Properties

Label 1-73-73.18-r0-0-0
Degree $1$
Conductor $73$
Sign $-0.510 + 0.859i$
Analytic cond. $0.339010$
Root an. cond. $0.339010$
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  + (0.173 + 0.984i)2-s + (−0.5 − 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.173 + 0.984i)5-s + (0.766 − 0.642i)6-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − 10-s + (−0.173 + 0.984i)11-s + (0.766 + 0.642i)12-s + (−0.766 + 0.642i)13-s + (−0.766 + 0.642i)14-s + (0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.5 − 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.173 + 0.984i)5-s + (0.766 − 0.642i)6-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − 10-s + (−0.173 + 0.984i)11-s + (0.766 + 0.642i)12-s + (−0.766 + 0.642i)13-s + (−0.766 + 0.642i)14-s + (0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.510 + 0.859i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ -0.510 + 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3557348560 + 0.6250182232i\)
\(L(\frac12)\) \(\approx\) \(0.3557348560 + 0.6250182232i\)
\(L(1)\) \(\approx\) \(0.6739214663 + 0.4864252549i\)
\(L(1)\) \(\approx\) \(0.6739214663 + 0.4864252549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.173 + 0.984i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 - T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−31.30312031788791778864921561636, −29.82834731482429894496833001472, −29.0605636504618734995192782251, −27.854985972553088964873925959553, −27.34251659609054626317035026448, −26.29475690649559001233034966623, −24.10440115705286357190971658332, −23.487503863278041765592314788042, −22.09763199425291080442066138971, −21.22436864077685849615795650846, −20.35415314266167174594386189286, −19.433956056540926745949385410893, −17.54760914949739177430463633734, −16.92592743546714489031566564702, −15.42169349758104473955810407928, −13.99169686444898034939585454545, −12.739816069064180927847409854025, −11.50769589295944305349447415100, −10.59325209515272509745420988432, −9.432975203736533999883079449974, −8.159421261754742112836788416845, −5.52445459390988731006146939852, −4.623437742376168022042134178635, −3.40408872390758693150334012283, −0.90125993454250221413200572783, 2.45090756919609866195854274518, 4.74473613431878477082665839741, 6.05491119018939840041586602289, 7.15650421530736864220398322942, 8.02258834179665450146046909262, 9.81409827422241514218718274638, 11.67298446349709786926394204391, 12.535316751613258029488813454642, 14.16260319472637335429433664718, 14.829080496855304110364356863091, 16.24671290625221515930022155626, 17.60655597694032032141850205973, 18.332492293028031686355569778925, 19.13850949572210608910869133870, 21.32680238430331936196260323077, 22.6845466717214385714810721397, 23.01836408666569613414288392376, 24.49788340388025475665138370339, 25.07414318944872711829814588821, 26.307668041084632697636347998248, 27.46278485680927461058182585910, 28.60564038843042138266674105748, 30.059833157400257370196137288462, 30.98202529258225572338208368094, 31.64012616840882419588227019295

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