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Properties

Label 2-273-7.2-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.990 + 0.139i$
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  + (−0.660 + 1.14i)2-s + (0.5 + 0.866i)3-s + (0.126 + 0.219i)4-s + (−1.01 + 1.75i)5-s − 1.32·6-s + (−2.38 − 1.14i)7-s − 2.97·8-s + (−0.499 + 0.866i)9-s + (−1.33 − 2.31i)10-s + (0.0340 + 0.0589i)11-s + (−0.126 + 0.219i)12-s − 13-s + (2.88 − 1.97i)14-s − 2.02·15-s + (1.71 − 2.96i)16-s + (3.02 + 5.23i)17-s + ⋯
L(s)  = 1  + (−0.467 + 0.809i)2-s + (0.288 + 0.499i)3-s + (0.0633 + 0.109i)4-s + (−0.452 + 0.784i)5-s − 0.539·6-s + (−0.901 − 0.432i)7-s − 1.05·8-s + (−0.166 + 0.288i)9-s + (−0.423 − 0.733i)10-s + (0.0102 + 0.0177i)11-s + (−0.0365 + 0.0633i)12-s − 0.277·13-s + (0.771 − 0.527i)14-s − 0.523·15-s + (0.428 − 0.742i)16-s + (0.733 + 1.26i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.990 + 0.139i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.990 + 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0522891 - 0.747796i\)
\(L(\frac12)\) \(\approx\) \(0.0522891 - 0.747796i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (0.660 - 1.14i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.01 - 1.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.0340 - 0.0589i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.02 - 5.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.67 + 4.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.77 - 4.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.38T + 29T^{2} \)
31 \( 1 + (-1.58 - 2.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.64 - 4.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.36T + 41T^{2} \)
43 \( 1 + 0.103T + 43T^{2} \)
47 \( 1 + (-2.03 + 3.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.97 - 6.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.44 - 5.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.02 + 6.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.33 - 4.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.522T + 71T^{2} \)
73 \( 1 + (-3.43 - 5.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.86 - 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (-2.48 + 4.31i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.09T + 97T^{2} \)
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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

−12.32288202210367514527597210985, −11.34339446409725327381798612371, −10.26913436693179865919270422487, −9.494637863960285871234972291021, −8.422431783800101909866752465289, −7.42731520195863950200130603603, −6.80863551870834428753370229213, −5.62806680762333810559843633817, −3.77047612747574846431466908349, −3.01358340933346204276852147067, 0.62797262760006554461863502597, 2.33844178423338720249672095284, 3.54533357777765772007650537986, 5.33924318672153719190741935410, 6.42977711773986612312506811892, 7.71734986724868969531994671496, 8.772641455016699188266632254275, 9.549110675650693048250550269172, 10.28843305978751543767735606317, 11.74224243974897389269767676860

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