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Properties

Label 2-273-91.16-c1-0-11
Degree $2$
Conductor $273$
Sign $-0.139 + 0.990i$
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  + (−0.5 + 0.866i)3-s − 2·4-s + (−2 − 1.73i)7-s + (−0.499 − 0.866i)9-s + (3 − 5.19i)11-s + (1 − 1.73i)12-s + (1 − 3.46i)13-s + 4·16-s − 6·17-s + (0.5 + 0.866i)19-s + (2.5 − 0.866i)21-s − 6·23-s + (2.5 + 4.33i)25-s + 0.999·27-s + (4 + 3.46i)28-s + (−3 − 5.19i)29-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s − 4-s + (−0.755 − 0.654i)7-s + (−0.166 − 0.288i)9-s + (0.904 − 1.56i)11-s + (0.288 − 0.499i)12-s + (0.277 − 0.960i)13-s + 16-s − 1.45·17-s + (0.114 + 0.198i)19-s + (0.545 − 0.188i)21-s − 1.25·23-s + (0.5 + 0.866i)25-s + 0.192·27-s + (0.755 + 0.654i)28-s + (−0.557 − 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.139 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.374122 - 0.430355i\)
\(L(\frac12)\) \(\approx\) \(0.374122 - 0.430355i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
13 \( 1 + (-1 + 3.46i)T \)
good2 \( 1 + 2T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.44611654484394909958756734412, −10.67916768698718852694132936098, −9.668201765387804100803087951291, −8.954477574557389266822481880619, −7.978813380158801354819627663218, −6.39530755092780552400217075952, −5.59153526927642284869914102854, −4.11700446424569758226342094974, −3.46599951897147741034859237712, −0.46472755962556151577273918472, 1.98576064178313417115910997199, 3.95070459286534361950933181856, 4.92613828564084320850381226945, 6.34412844073955488095690382710, 7.02912017143054913870090716736, 8.574753939850581975155307663895, 9.238726619268364511640779083065, 10.03338017554999544962100238572, 11.43451544201108740016613951368, 12.43910921535886099413207746231

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