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Properties

Label 1-273-273.80-r1-0-0
Degree $1$
Conductor $273$
Sign $0.100 - 0.994i$
Analytic cond. $29.3379$
Root an. cond. $29.3379$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s − i·8-s + 10-s − i·11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + i·19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s − i·8-s + 10-s − i·11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + i·19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ 0.100 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.868702396 - 2.592771865i\)
\(L(\frac12)\) \(\approx\) \(2.868702396 - 2.592771865i\)
\(L(1)\) \(\approx\) \(1.895033399 - 0.8522740537i\)
\(L(1)\) \(\approx\) \(1.895033399 - 0.8522740537i\)

Euler product

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

−25.80446708932654647693909341494, −24.70744530314254500486948810464, −23.90363051956152312342991778165, −23.08294214486952908624980706718, −21.96933622900060921689543529521, −21.39677196855784119602169186307, −20.47568137976394670053365042906, −19.6056919468134789155503780469, −17.828633129109673652402081174601, −17.4167175360859847163449950383, −16.36676903227321874392496985705, −15.407232915102413955519278824993, −14.471176482790033792629714606629, −13.55716496548867467646395287306, −12.74184188162082192408353009416, −11.97348055222287366255885706760, −10.57948952900994810281831423004, −9.43372965907410280497856821654, −8.29901847814622723659650144934, −7.13399821423469562081335317511, −6.12239646825979226048761690154, −5.15159806153537778926053131862, −4.25906332805877549553704869622, −2.782591820315028072588106537220, −1.57533493006363526631362010773, 0.92332999332860958946714688452, 2.34932483927114060945175230178, 3.215648524234164141976847882325, 4.54621337916644065875350981656, 5.85959286198268077424986805380, 6.31172738739639763160573980088, 7.8117955638848116619902167107, 9.42030175896734844762027512781, 10.23477391292583670437245656938, 11.152550092097868727571328061948, 12.122707149222101620254188754129, 13.25592969505264651417343221769, 14.04332906147355416869053329591, 14.60952416146965299109982135577, 15.91455110760345606729513190693, 16.80952129307369137465061777706, 18.28289554480045929856840367810, 18.82820929439378373126466429784, 19.96818141803840240836684267656, 21.12654605945714852106170364060, 21.43840960860212160833469299408, 22.624123147478025709141083104, 23.07791844573888249829879190582, 24.5639092038204155580284867742, 24.84183414526034125516176557854

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