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Properties

Label 2-273-273.95-c0-0-0
Degree $2$
Conductor $273$
Sign $0.543 - 0.839i$
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  + (0.5 + 0.866i)3-s − 4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)12-s + (0.5 − 0.866i)13-s + 16-s + (−1.5 − 0.866i)19-s + (−0.499 + 0.866i)21-s + (0.5 − 0.866i)25-s − 0.999·27-s + (−0.5 − 0.866i)28-s + (0.499 − 0.866i)36-s − 1.73i·37-s + 0.999·39-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s − 4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)12-s + (0.5 − 0.866i)13-s + 16-s + (−1.5 − 0.866i)19-s + (−0.499 + 0.866i)21-s + (0.5 − 0.866i)25-s − 0.999·27-s + (−0.5 − 0.866i)28-s + (0.499 − 0.866i)36-s − 1.73i·37-s + 0.999·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.543 - 0.839i$
Analytic conductor: \(0.136244\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :0),\ 0.543 - 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7431114630\)
\(L(\frac12)\) \(\approx\) \(0.7431114630\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
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 \( 1 + T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
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 \( 1 + T^{2} \)
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

−12.46799419406341853253916434073, −11.06044224702581829300215777349, −10.36961570628701526227411392803, −9.184746018849948894942346576850, −8.676571157915595226815636698647, −7.906844754017836480402007131656, −5.95423315817357622615728553006, −4.96317471180409279700529314242, −4.06639766111190915217367692288, −2.63246840570324538629877096217, 1.55532597752984363062862192735, 3.58034364638658197482065533815, 4.57745593534112865804498562797, 6.11878125073448870038652155030, 7.21151721168130198249337962751, 8.259617039379478357178920023381, 8.834935623988358616706225044344, 9.986493602966646755640094683693, 11.07613514157848440107508738172, 12.21013744596379579172905206964

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