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Properties

Label 2-273-273.269-c1-0-12
Degree $2$
Conductor $273$
Sign $0.662 + 0.748i$
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  − 0.843i·2-s + (−1.29 − 1.14i)3-s + 1.28·4-s + (1.06 + 1.84i)5-s + (−0.968 + 1.09i)6-s + (2.34 + 1.22i)7-s − 2.77i·8-s + (0.360 + 2.97i)9-s + (1.55 − 0.896i)10-s + (0.855 − 0.493i)11-s + (−1.67 − 1.48i)12-s + (3.08 + 1.86i)13-s + (1.03 − 1.97i)14-s + (0.737 − 3.60i)15-s + 0.240·16-s − 1.84·17-s + ⋯
L(s)  = 1  − 0.596i·2-s + (−0.748 − 0.663i)3-s + 0.644·4-s + (0.475 + 0.823i)5-s + (−0.395 + 0.446i)6-s + (0.886 + 0.463i)7-s − 0.980i·8-s + (0.120 + 0.992i)9-s + (0.490 − 0.283i)10-s + (0.257 − 0.148i)11-s + (−0.482 − 0.427i)12-s + (0.855 + 0.518i)13-s + (0.276 − 0.528i)14-s + (0.190 − 0.931i)15-s + 0.0600·16-s − 0.446·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.662 + 0.748i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.662 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28203 - 0.577307i\)
\(L(\frac12)\) \(\approx\) \(1.28203 - 0.577307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.29 + 1.14i)T \)
7 \( 1 + (-2.34 - 1.22i)T \)
13 \( 1 + (-3.08 - 1.86i)T \)
good2 \( 1 + 0.843iT - 2T^{2} \)
5 \( 1 + (-1.06 - 1.84i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.855 + 0.493i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.84T + 17T^{2} \)
19 \( 1 + (6.19 + 3.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.62iT - 23T^{2} \)
29 \( 1 + (-0.488 - 0.281i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.23 + 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.29T + 37T^{2} \)
41 \( 1 + (4.93 - 8.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.214 + 0.370i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.40 - 11.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.59 + 3.80i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 + (-1.92 - 1.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.22 - 2.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.48 + 1.43i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.31 + 4.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.27 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 5.18T + 89T^{2} \)
97 \( 1 + (0.172 - 0.0997i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.44907336503623679415951471069, −11.08941032151908809112279608562, −10.44332739049861641988152903513, −8.956778352051963023107613535232, −7.69752997575354053977430059159, −6.48424628868315668807842514080, −6.18988025063443475796368448135, −4.50531439529721313957569188175, −2.61531336372782118178279641888, −1.64864084957145442537996846562, 1.61220772175311403545631515480, 3.95638260628068660056873827855, 5.14385298629465659950365293036, 5.88290531311987464185045135829, 6.91121612424261789547969299712, 8.209902666792774095966310444501, 9.016248859002807734368540961160, 10.46543042964214371869578243225, 10.91036570158931466447020793073, 11.90838122156871311620290471047

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