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Properties

Label 2-273-273.233-c2-0-20
Degree $2$
Conductor $273$
Sign $-0.914 - 0.403i$
Analytic cond. $7.43871$
Root an. cond. $2.72740$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.759 + 1.31i)2-s + (2.79 + 1.09i)3-s + (0.845 + 1.46i)4-s + (−0.681 + 1.18i)5-s + (−3.56 + 2.84i)6-s + (−0.736 + 6.96i)7-s − 8.64·8-s + (6.59 + 6.11i)9-s + (−1.03 − 1.79i)10-s + (−2.96 − 5.12i)11-s + (0.756 + 5.01i)12-s − 13·13-s + (−8.60 − 6.25i)14-s + (−3.19 + 2.54i)15-s + (3.19 − 5.52i)16-s + (8.60 − 4.96i)17-s + ⋯
L(s)  = 1  + (−0.379 + 0.658i)2-s + (0.930 + 0.365i)3-s + (0.211 + 0.365i)4-s + (−0.136 + 0.236i)5-s + (−0.594 + 0.473i)6-s + (−0.105 + 0.994i)7-s − 1.08·8-s + (0.733 + 0.679i)9-s + (−0.103 − 0.179i)10-s + (−0.269 − 0.466i)11-s + (0.0630 + 0.417i)12-s − 13-s + (−0.614 − 0.447i)14-s + (−0.213 + 0.169i)15-s + (0.199 − 0.345i)16-s + (0.506 − 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.403i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.914 - 0.403i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ -0.914 - 0.403i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.330554 + 1.56693i\)
\(L(\frac12)\) \(\approx\) \(0.330554 + 1.56693i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.79 - 1.09i)T \)
7 \( 1 + (0.736 - 6.96i)T \)
13 \( 1 + 13T \)
good2 \( 1 + (0.759 - 1.31i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + (0.681 - 1.18i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (2.96 + 5.12i)T + (-60.5 + 104. i)T^{2} \)
17 \( 1 + (-8.60 + 4.96i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-7.13 - 4.11i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-2.23 - 1.29i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 21.1iT - 841T^{2} \)
31 \( 1 + (-9.34 + 5.39i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-28.5 - 16.4i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 16.9T + 1.68e3T^{2} \)
43 \( 1 + 3.30T + 1.84e3T^{2} \)
47 \( 1 + (32.9 - 57.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (58.6 - 33.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-8.56 - 14.8i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (9.60 - 16.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-88.3 + 50.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 27.8T + 5.04e3T^{2} \)
73 \( 1 + (-87.1 + 50.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-28.4 + 49.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 104.T + 6.88e3T^{2} \)
89 \( 1 + (43.7 - 75.7i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 27.8iT - 9.40e3T^{2} \)
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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

−12.14750028720117048181131936273, −11.15366000644928253393928535516, −9.708046028050249767513788834829, −9.177439324869134777615452855193, −8.056341080846176677769260242911, −7.59142494157357832511295938405, −6.31223872218118384135619645263, −5.03497975335570590202821060651, −3.31761489615501926865465966278, −2.52776880852218163536268424801, 0.814999746596776804421839893438, 2.21175625546058801077049749773, 3.43202079091827416116660540331, 4.87086962735702022823627193136, 6.59124703074339632462320760681, 7.44199167986170510226377102708, 8.462624375133010895662427108366, 9.691552226683707370936390430799, 10.04158853783399390292095537525, 11.14972885783222986297856514961

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