Properties

Label 1-273-273.215-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: $\chi_{273} (215, \cdot )$
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

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

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