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Properties

Label 2-273-273.44-c1-0-4
Degree $2$
Conductor $273$
Sign $0.254 - 0.967i$
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  + (−0.866 − 1.5i)3-s + (−1.73 + i)4-s + (0.5 + 2.59i)7-s + (−1.5 + 2.59i)9-s + (3 + 1.73i)12-s + (0.866 + 3.5i)13-s + (1.99 − 3.46i)16-s + (2.23 + 8.33i)19-s + (3.46 − 3i)21-s + (−4.33 + 2.5i)25-s + 5.19·27-s + (−3.46 − 4i)28-s + (−1.13 − 0.303i)31-s − 6i·36-s + (−7.59 + 2.03i)37-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (−0.866 + 0.5i)4-s + (0.188 + 0.981i)7-s + (−0.5 + 0.866i)9-s + (0.866 + 0.499i)12-s + (0.240 + 0.970i)13-s + (0.499 − 0.866i)16-s + (0.512 + 1.91i)19-s + (0.755 − 0.654i)21-s + (−0.866 + 0.5i)25-s + 1.00·27-s + (−0.654 − 0.755i)28-s + (−0.203 − 0.0545i)31-s − i·36-s + (−1.24 + 0.334i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.254 - 0.967i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.562560 + 0.433757i\)
\(L(\frac12)\) \(\approx\) \(0.562560 + 0.433757i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
13 \( 1 + (-0.866 - 3.5i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
5 \( 1 + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.23 - 8.33i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.13 + 0.303i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (7.59 - 2.03i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 41iT^{2} \)
43 \( 1 + 12.1iT - 43T^{2} \)
47 \( 1 + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.46 - 6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-15.7 - 4.23i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 71iT^{2} \)
73 \( 1 + (1.57 - 5.86i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.06 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-13.9 + 13.9i)T - 97iT^{2} \)
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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

−12.05051161249350788306710884163, −11.64926956441607718259106889269, −10.17070326579708179768636210117, −9.010701548524652366668296916865, −8.253363710990125608288182104209, −7.31777463497987380024673549703, −5.97467809956692366007062704781, −5.16593297280871508394435663282, −3.66550402670282311520218583431, −1.86332951529759131415440609928, 0.61155138404384136126541772739, 3.46092025716659151253838750154, 4.58114025526944434769903628398, 5.31382518620491011752720717662, 6.53858930006306273976638313649, 7.964564868613916924223372206597, 9.114631809677071524064309820320, 9.892222426748207199946685520760, 10.68158275237239587336183313766, 11.35468222750416101592585456704

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