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Properties

Label 2-273-273.173-c1-0-24
Degree $2$
Conductor $273$
Sign $-0.0614 + 0.998i$
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  − 1.73i·3-s + (1 − 1.73i)4-s + (2.5 − 0.866i)7-s − 2.99·9-s + (−2.99 − 1.73i)12-s + (−1 + 3.46i)13-s + (−1.99 − 3.46i)16-s + 19-s + (−1.49 − 4.33i)21-s + (−2.5 − 4.33i)25-s + 5.19i·27-s + (1.00 − 5.19i)28-s + (3.5 + 6.06i)31-s + (−2.99 + 5.19i)36-s + (4.5 − 2.59i)37-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.5 − 0.866i)4-s + (0.944 − 0.327i)7-s − 0.999·9-s + (−0.866 − 0.499i)12-s + (−0.277 + 0.960i)13-s + (−0.499 − 0.866i)16-s + 0.229·19-s + (−0.327 − 0.944i)21-s + (−0.5 − 0.866i)25-s + 0.999i·27-s + (0.188 − 0.981i)28-s + (0.628 + 1.08i)31-s + (−0.499 + 0.866i)36-s + (0.739 − 0.427i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0614 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0614 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0614 + 0.998i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0614 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.990260 - 1.05310i\)
\(L(\frac12)\) \(\approx\) \(0.990260 - 1.05310i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
7 \( 1 + (-2.5 + 0.866i)T \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.5 + 2.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 15.5iT - 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.5 + 16.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.59274146321204090495889371705, −10.98105627396780032222689239593, −9.845918452361718278981875229179, −8.636590635276103521696480366800, −7.56917064314693877112065198082, −6.75189438300436116532998695716, −5.77282430805161703084340158019, −4.57562646983351975733300110648, −2.43711247051589319164585414871, −1.26970395328331028953188725527, 2.49036549984100470823466325572, 3.70173792585429273771924206217, 4.89222525149688574510108660161, 5.96524646444371478154392917413, 7.58221246124543804927297475977, 8.235021709138087269948939634017, 9.257672160291920509980622102825, 10.38807944101643464127052967140, 11.29987303327754058531531549449, 11.84279102773458749891848718418

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