Properties

Label 2-273-91.24-c1-0-10
Degree $2$
Conductor $273$
Sign $0.882 - 0.470i$
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  + (1.64 − 0.441i)2-s + i·3-s + (0.784 − 0.453i)4-s + (1.35 + 0.363i)5-s + (0.441 + 1.64i)6-s + (0.501 + 2.59i)7-s + (−1.31 + 1.31i)8-s − 9-s + 2.39·10-s + (1.07 − 1.07i)11-s + (0.453 + 0.784i)12-s + (1.37 − 3.33i)13-s + (1.97 + 4.05i)14-s + (−0.363 + 1.35i)15-s + (−2.49 + 4.32i)16-s + (−2.58 − 4.47i)17-s + ⋯
L(s)  = 1  + (1.16 − 0.311i)2-s + 0.577i·3-s + (0.392 − 0.226i)4-s + (0.607 + 0.162i)5-s + (0.180 + 0.672i)6-s + (0.189 + 0.981i)7-s + (−0.466 + 0.466i)8-s − 0.333·9-s + 0.758·10-s + (0.323 − 0.323i)11-s + (0.130 + 0.226i)12-s + (0.380 − 0.924i)13-s + (0.527 + 1.08i)14-s + (−0.0939 + 0.350i)15-s + (−0.623 + 1.08i)16-s + (−0.626 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.882 - 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20939 + 0.551758i\)
\(L(\frac12)\) \(\approx\) \(2.20939 + 0.551758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-0.501 - 2.59i)T \)
13 \( 1 + (-1.37 + 3.33i)T \)
good2 \( 1 + (-1.64 + 0.441i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.35 - 0.363i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.07 + 1.07i)T - 11iT^{2} \)
17 \( 1 + (2.58 + 4.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.42 + 3.42i)T - 19iT^{2} \)
23 \( 1 + (-1.86 - 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.744 + 1.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.506 - 1.89i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.45 - 9.16i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.34 + 1.70i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.27 + 4.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.66 + 6.20i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.87 + 3.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.19 + 8.17i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 3.23iT - 61T^{2} \)
67 \( 1 + (-9.21 - 9.21i)T + 67iT^{2} \)
71 \( 1 + (2.71 - 0.726i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (14.2 - 3.82i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.67 - 6.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0684 - 0.0684i)T - 83iT^{2} \)
89 \( 1 + (-10.7 + 2.88i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.39 - 12.6i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.71920810592766783683107306965, −11.56260470355742475729553618067, −10.18741667940242982682207212784, −9.158959844006378269391162036745, −8.381739308859411948499434537695, −6.57048923987942053478172227722, −5.46731234857583119391220391201, −4.96104038877219376644941513668, −3.43109959976787521840668930443, −2.50815529289745973786034629718, 1.64876040277807984885764367389, 3.63379636077998310391046737243, 4.57470699710631177441555528809, 5.85279440431510080297111313507, 6.59412132510963965435236575315, 7.56513790483488468799770455738, 8.962486240237948651944812880648, 9.937737038336167765797655651905, 11.16445652424807026731372701249, 12.14250643279588461940068108197

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