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Properties

Label 1-273-273.251-r0-0-0
Degree $1$
Conductor $273$
Sign $0.859 + 0.511i$
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 − 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 + (0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)29-s + 31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 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 + (0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)29-s + 31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5135586810 + 0.1411430276i\)
\(L(\frac12)\) \(\approx\) \(0.5135586810 + 0.1411430276i\)
\(L(1)\) \(\approx\) \(0.6017569688 - 0.1106713302i\)
\(L(1)\) \(\approx\) \(0.6017569688 - 0.1106713302i\)

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 \( 1 - T \)
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 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (0.5 + 0.866i)T \)
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

−25.68742526337837103731446763426, −24.71050582838792929656155773860, −23.92488580526780790334517949458, −23.014379407913720670134164724662, −22.536355083267952106945338240918, −20.89181875838214014452429169555, −19.88487560433457149669605183226, −19.144972334902855619750126542837, −18.17792708467331233506669645798, −17.36131001200862238574205206714, −16.23737440911609811029260550044, −15.49827509263361018037118969066, −14.88288926042333681162439499164, −13.66261627885014946620328211940, −12.5682211979298526312005121287, −11.32005221264457771289512425570, −10.34945419915589699018800871399, −9.18970370506593350498551802217, −8.25086935785477590649907495580, −7.31689178239205641533662377878, −6.56448757484555130448935096015, −4.98247020960558981482414439596, −4.309705619748458653653835210940, −2.47414651560401942033751270450, −0.48568130809205055328353223233, 1.26075068560597245070512962001, 2.88922937073979863441574210905, 3.76427415971740557466556976873, 4.88601797585690217608325128539, 6.580957482136193125478310664187, 8.07661873877982225269706287625, 8.33725822382200731714908432367, 9.74285100837815344316971367338, 10.86052471668418423962618499223, 11.432451815137126145912967350660, 12.53008019085968112545936420, 13.29447930960984602448931406384, 14.61040157287384543112300435491, 15.82110723142753711230007846140, 16.63152560537780410165393622313, 17.680843095771461526185621196701, 18.79575934345764905402024963764, 19.30891196793585163468867828549, 20.161200291453375449721970949169, 21.1857907293435141272585224398, 21.909896900796224774348288464329, 23.074648991290827116104097345334, 23.73746674432040669985940435936, 25.00021480671537741216576350093, 26.17862107903557069826704929446

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