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Properties

Label 2-273-91.62-c2-0-33
Degree $2$
Conductor $273$
Sign $0.102 + 0.994i$
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  + (3.01 − 1.74i)2-s + (−1.5 + 0.866i)3-s + (4.07 − 7.06i)4-s + 5.35·5-s + (−3.01 + 5.23i)6-s + (−6.87 − 1.33i)7-s − 14.5i·8-s + (1.5 − 2.59i)9-s + (16.1 − 9.33i)10-s + (14.7 − 8.52i)11-s + 14.1i·12-s + (−3.94 − 12.3i)13-s + (−23.0 + 7.94i)14-s + (−8.02 + 4.63i)15-s + (−8.96 − 15.5i)16-s + (8.84 + 5.10i)17-s + ⋯
L(s)  = 1  + (1.50 − 0.871i)2-s + (−0.5 + 0.288i)3-s + (1.01 − 1.76i)4-s + 1.07·5-s + (−0.503 + 0.871i)6-s + (−0.981 − 0.190i)7-s − 1.81i·8-s + (0.166 − 0.288i)9-s + (1.61 − 0.933i)10-s + (1.34 − 0.775i)11-s + 1.17i·12-s + (−0.303 − 0.952i)13-s + (−1.64 + 0.567i)14-s + (−0.535 + 0.308i)15-s + (−0.560 − 0.971i)16-s + (0.520 + 0.300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.102 + 0.994i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ 0.102 + 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.58486 - 2.33110i\)
\(L(\frac12)\) \(\approx\) \(2.58486 - 2.33110i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (6.87 + 1.33i)T \)
13 \( 1 + (3.94 + 12.3i)T \)
good2 \( 1 + (-3.01 + 1.74i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 5.35T + 25T^{2} \)
11 \( 1 + (-14.7 + 8.52i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (-8.84 - 5.10i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (4.19 - 7.27i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.33 - 14.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-16.5 - 28.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 56.0T + 961T^{2} \)
37 \( 1 + (23.4 - 13.5i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (0.675 + 1.17i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (2.44 - 4.23i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 20.3T + 2.20e3T^{2} \)
53 \( 1 + 3.47T + 2.80e3T^{2} \)
59 \( 1 + (37.2 - 64.5i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-49.8 - 28.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (31.8 - 18.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-62.4 - 36.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 - 77.4T + 5.32e3T^{2} \)
79 \( 1 + 130.T + 6.24e3T^{2} \)
83 \( 1 + 86.4T + 6.88e3T^{2} \)
89 \( 1 + (8.88 + 15.3i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-91.8 + 159. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.64894792514179336779376981593, −10.62772242323589664110688343306, −10.04411255404657392984186852454, −9.065543776917293236842312695859, −6.85328797120741368841755957378, −5.87930797231948900640991192152, −5.45434695794187777967330838971, −3.89439671739774579895758463501, −3.09493321510398479750207692744, −1.38068714067131814982422870597, 2.18308202516816748266630400368, 3.79865162943710276341357630767, 4.93682547322692880734384158693, 5.99702506918423833048375649232, 6.61073093049658050457497272341, 7.23559536994486931735044272607, 9.107791805385879117624639479988, 9.896039267780539516366571183763, 11.46659557247324650832845260016, 12.37117040814348922113047587524

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