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Properties

Label 2-273-91.88-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.555 + 0.831i$
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  + (0.754 + 0.435i)2-s − 3-s + (−0.620 − 1.07i)4-s + (−1.34 + 0.773i)5-s + (−0.754 − 0.435i)6-s + (−2.48 − 0.905i)7-s − 2.82i·8-s + 9-s − 1.34·10-s − 3.61i·11-s + (0.620 + 1.07i)12-s + (−3.37 + 1.26i)13-s + (−1.48 − 1.76i)14-s + (1.34 − 0.773i)15-s + (−0.00935 + 0.0161i)16-s + (−2.31 − 4.01i)17-s + ⋯
L(s)  = 1  + (0.533 + 0.308i)2-s − 0.577·3-s + (−0.310 − 0.537i)4-s + (−0.599 + 0.345i)5-s + (−0.308 − 0.177i)6-s + (−0.939 − 0.342i)7-s − 0.998i·8-s + 0.333·9-s − 0.426·10-s − 1.09i·11-s + (0.179 + 0.310i)12-s + (−0.936 + 0.351i)13-s + (−0.396 − 0.472i)14-s + (0.345 − 0.199i)15-s + (−0.00233 + 0.00404i)16-s + (−0.561 − 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.261087 - 0.488597i\)
\(L(\frac12)\) \(\approx\) \(0.261087 - 0.488597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (2.48 + 0.905i)T \)
13 \( 1 + (3.37 - 1.26i)T \)
good2 \( 1 + (-0.754 - 0.435i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.34 - 0.773i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 3.61iT - 11T^{2} \)
17 \( 1 + (2.31 + 4.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 0.586iT - 19T^{2} \)
23 \( 1 + (-1.14 + 1.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.07 - 1.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.99 - 5.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.03 + 2.90i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.60 - 0.926i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.93 + 6.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.10 - 1.21i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.49 + 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.68 + 2.70i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 9.70T + 61T^{2} \)
67 \( 1 - 1.92iT - 67T^{2} \)
71 \( 1 + (9.14 + 5.28i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.96 - 1.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.78 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.45iT - 83T^{2} \)
89 \( 1 + (10.9 + 6.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.3 + 8.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.63804280827102308800831122194, −10.64507135775923200896925036404, −9.843105787228752866624417508282, −8.810717723835789926104624390162, −7.13072914359877220418509833639, −6.62700767456277338800546377117, −5.45819992657176825080036135680, −4.42803109771300135847083734947, −3.20121594965749253047048802135, −0.37337177802891473440980066201, 2.57816378114573229121169090667, 4.04598579172575945560301694379, 4.79629706585338224155184842862, 6.09672890518609426656577627292, 7.34938765627502386489587757308, 8.310042982568283342239487491046, 9.502449067339870795141295972688, 10.39884776111465129461368804507, 11.79818550957367092628216417802, 12.21172034326999376225932882448

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