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Properties

Label 2-273-273.212-c0-0-0
Degree $2$
Conductor $273$
Sign $0.564 + 0.825i$
Analytic cond. $0.136244$
Root an. cond. $0.369113$
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  − 3-s + (0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s + 9-s + (−0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 1.73i·19-s + (0.5 + 0.866i)21-s + (0.5 + 0.866i)25-s − 27-s − 0.999·28-s + (0.5 − 0.866i)36-s + (−1.5 + 0.866i)37-s + (−0.5 + 0.866i)39-s + ⋯
L(s)  = 1  − 3-s + (0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s + 9-s + (−0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 1.73i·19-s + (0.5 + 0.866i)21-s + (0.5 + 0.866i)25-s − 27-s − 0.999·28-s + (0.5 − 0.866i)36-s + (−1.5 + 0.866i)37-s + (−0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(0.136244\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :0),\ 0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6168446239\)
\(L(\frac12)\) \(\approx\) \(0.6168446239\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.87490388375215911035924395721, −10.77707517022733189925787755119, −10.41696182587682520291866098780, −9.568451186236934005366202867373, −7.83973197499347162017517243446, −6.79833414770465699130613908780, −6.00832376685068544816406181762, −5.08950355497316490780574246973, −3.59200892656335830214746979321, −1.32217805949075965076820073204, 2.38374102043130098665145034659, 3.93805820588745944624488189519, 5.26080347827440286494183730227, 6.54316899649752673161633725075, 7.00229180965700669090160272860, 8.510731366976896533047150480091, 9.359248616527200161023906761317, 10.73729017302974224375945052043, 11.44025993680020983483871662485, 12.21478609160988315795871640522

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