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Properties

Label 1-273-273.38-r0-0-0
Degree $1$
Conductor $273$
Sign $0.605 - 0.795i$
Analytic cond. $1.26780$
Root an. cond. $1.26780$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7015429952 - 0.3477739833i\)
\(L(\frac12)\) \(\approx\) \(0.7015429952 - 0.3477739833i\)
\(L(1)\) \(\approx\) \(0.7822482443 + 0.008901401302i\)
\(L(1)\) \(\approx\) \(0.7822482443 + 0.008901401302i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−25.77765037167928245212346115326, −25.578011661623474132096281188122, −23.87052690681569401365732381502, −22.84841086629472154639469898747, −21.98566435528974241066444208338, −21.34065752989929390140091747010, −20.32269689454227394572538696728, −19.393458487292944773802330817030, −18.560900233358172265909634810411, −17.65168857414060706409325733709, −17.15781967815146446907485719301, −15.59996665140325730362369511131, −14.65115737034036915716063128481, −13.37896244880688135641247750565, −12.78033738332933221862936204416, −11.422194476112654752033594974692, −10.69058413021996957848265854868, −9.85210425106006401960381506817, −8.91035140737686497765283297576, −7.6179819284093525964634757003, −6.714594588901156391019465947885, −5.14637240180852063909699803613, −3.81956058969153584076369973580, −2.636372646877967619573544222105, −1.72712557476740683689033704485, 0.626909943364665538059155313429, 2.16715891755494195775476840467, 4.16616417519851315072101885244, 5.33677325544248241227981045640, 6.02399604777698606589089525273, 7.32854479158215941909275240436, 8.43532352053861681094553953911, 9.09512701441921690085803247094, 10.13485625070778234013222011629, 11.198492944799412445758586190636, 12.76091925335522470446153382112, 13.54381935527992524143146407099, 14.46897612890482409727716567646, 15.6218491967033064608673253363, 16.534058572652561426929849997756, 16.99365617189020335248204357909, 18.22884777802095354007505591147, 18.828959691671442994623031171696, 20.09415779714664850877600890761, 20.90405145826409872613050384420, 22.065955683403736894685073674322, 23.11422556555935184466510021612, 24.09669518726907028094932382410, 24.68311390087944269242569376553, 25.4508949866986026285889277867

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