Properties

Label 2-273-273.227-c0-0-0
Degree $2$
Conductor $273$
Sign $0.906 + 0.421i$
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  + (−0.866 + 0.5i)3-s i·4-s + (0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (0.5 + 0.866i)12-s + (0.866 + 0.5i)13-s − 16-s + (−0.5 + 0.133i)19-s + (−0.499 + 0.866i)21-s + (−0.866 − 0.5i)25-s + 0.999i·27-s + (−0.5 − 0.866i)28-s + (−1.36 + 0.366i)31-s + (−0.866 − 0.499i)36-s + (1.36 + 1.36i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s i·4-s + (0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (0.5 + 0.866i)12-s + (0.866 + 0.5i)13-s − 16-s + (−0.5 + 0.133i)19-s + (−0.499 + 0.866i)21-s + (−0.866 − 0.5i)25-s + 0.999i·27-s + (−0.5 − 0.866i)28-s + (−1.36 + 0.366i)31-s + (−0.866 − 0.499i)36-s + (1.36 + 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.906 + 0.421i$
Analytic conductor: \(0.136244\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :0),\ 0.906 + 0.421i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6415837040\)
\(L(\frac12)\) \(\approx\) \(0.6415837040\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + iT^{2} \)
5 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
41 \( 1 + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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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

−11.56638594159037571451617487971, −11.17940899166395310978364890070, −10.31026669631553263824514617642, −9.523346298909165842596630898267, −8.308660176718113188897170199922, −6.82387818687788767839684934779, −5.97423186250830402451939335800, −4.93757266341048901716305037813, −4.03834894780255131311491278359, −1.50214803518883393025586550229, 2.07231612362859248253041835300, 3.87042298282541794138872455744, 5.17358557078102877837631555532, 6.19970708999153305762628923764, 7.46453818625146142633300934425, 8.107211141571774721641999490708, 9.151878976193034306692145014547, 10.77008989429976638105072027551, 11.34984707967465724632257761781, 12.14283645568478792849058137253

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