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Properties

Label 2-273-91.74-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.712 - 0.701i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.71·2-s + (0.5 + 0.866i)3-s + 0.941·4-s + (1.22 + 2.12i)5-s + (−0.857 − 1.48i)6-s + (−2.38 + 1.14i)7-s + 1.81·8-s + (−0.499 + 0.866i)9-s + (−2.10 − 3.64i)10-s + (0.519 + 0.899i)11-s + (0.470 + 0.815i)12-s + (−3.36 − 1.29i)13-s + (4.09 − 1.95i)14-s + (−1.22 + 2.12i)15-s − 4.99·16-s + 3.01·17-s + ⋯
L(s)  = 1  − 1.21·2-s + (0.288 + 0.499i)3-s + 0.470·4-s + (0.549 + 0.951i)5-s + (−0.350 − 0.606i)6-s + (−0.902 + 0.431i)7-s + 0.641·8-s + (−0.166 + 0.288i)9-s + (−0.666 − 1.15i)10-s + (0.156 + 0.271i)11-s + (0.135 + 0.235i)12-s + (−0.932 − 0.360i)13-s + (1.09 − 0.523i)14-s + (−0.317 + 0.549i)15-s − 1.24·16-s + 0.732·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209121 + 0.510121i\)
\(L(\frac12)\) \(\approx\) \(0.209121 + 0.510121i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.38 - 1.14i)T \)
13 \( 1 + (3.36 + 1.29i)T \)
good2 \( 1 + 1.71T + 2T^{2} \)
5 \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.519 - 0.899i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.01T + 17T^{2} \)
19 \( 1 + (1.59 - 2.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 + (4.01 - 6.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 + (2.54 - 4.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.21 + 5.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.88 - 8.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.90 + 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 + (-1.30 + 2.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.61 - 9.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.63 + 2.82i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.50 + 13.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.211 + 0.366i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.34T + 83T^{2} \)
89 \( 1 - 5.30T + 89T^{2} \)
97 \( 1 + (-2.92 - 5.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.11720521442764320257052450724, −10.70693345463425579129261561935, −10.10520356596808849294182661529, −9.614492934094612534847755805896, −8.671059553832105877154075923504, −7.53802917041844690449838989559, −6.61475826293087671362967154346, −5.28790933877756475325226722467, −3.52405211231960862557030281215, −2.21272095362163424881870186958, 0.59230488831685253035940714283, 2.16347565537627629671393030801, 4.13822943508873546952885738798, 5.66315120270283369762512748660, 6.96311224405383454221889403027, 7.81627605397807259387924687424, 8.842827608204304196048391331645, 9.560820567244475939590616711160, 10.06879359483769413911756680217, 11.45770961972059028375232261917

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