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Properties

Label 2-273-91.5-c1-0-14
Degree $2$
Conductor $273$
Sign $-0.468 + 0.883i$
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  + (−0.458 − 1.71i)2-s + (0.866 − 0.5i)3-s + (−0.987 + 0.570i)4-s + (2.18 − 0.586i)5-s + (−1.25 − 1.25i)6-s + (−0.537 + 2.59i)7-s + (−1.07 − 1.07i)8-s + (0.499 − 0.866i)9-s + (−2.00 − 3.47i)10-s + (0.937 − 3.49i)11-s + (−0.570 + 0.987i)12-s + (1.56 − 3.24i)13-s + (4.68 − 0.268i)14-s + (1.60 − 1.60i)15-s + (−2.49 + 4.31i)16-s + (2.54 + 4.40i)17-s + ⋯
L(s)  = 1  + (−0.324 − 1.21i)2-s + (0.499 − 0.288i)3-s + (−0.493 + 0.285i)4-s + (0.979 − 0.262i)5-s + (−0.511 − 0.511i)6-s + (−0.202 + 0.979i)7-s + (−0.380 − 0.380i)8-s + (0.166 − 0.288i)9-s + (−0.635 − 1.09i)10-s + (0.282 − 1.05i)11-s + (−0.164 + 0.285i)12-s + (0.435 − 0.900i)13-s + (1.25 − 0.0718i)14-s + (0.413 − 0.413i)15-s + (−0.622 + 1.07i)16-s + (0.616 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.468 + 0.883i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.748556 - 1.24415i\)
\(L(\frac12)\) \(\approx\) \(0.748556 - 1.24415i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.537 - 2.59i)T \)
13 \( 1 + (-1.56 + 3.24i)T \)
good2 \( 1 + (0.458 + 1.71i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-2.18 + 0.586i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.937 + 3.49i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.54 - 4.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.91 - 0.781i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.54 + 1.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.91T + 29T^{2} \)
31 \( 1 + (0.708 - 2.64i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (5.36 - 1.43i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.23 - 5.23i)T + 41iT^{2} \)
43 \( 1 - 5.01iT - 43T^{2} \)
47 \( 1 + (1.55 + 5.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.73 - 9.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.22 + 0.328i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-8.11 - 4.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.63 - 2.04i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.84 + 1.84i)T - 71iT^{2} \)
73 \( 1 + (-9.26 - 2.48i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.0621 + 0.107i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.88 + 9.88i)T + 83iT^{2} \)
89 \( 1 + (4.71 + 17.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.0 + 10.0i)T + 97iT^{2} \)
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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

−11.60104826775634912235509211192, −10.51829992623836349952464747928, −9.848375480136838301636806200743, −8.812235493270513057677745139868, −8.323187402335929254030646088975, −6.29992438247415041646034561478, −5.72786420593569808796086114340, −3.61174350911166732130500041561, −2.58181242286022029376840195740, −1.40295993702029833739477317081, 2.22186452874432823831813629322, 4.01122276323729201461751533517, 5.34632179165002680494735652237, 6.63832016828239883424439392968, 7.11984869114330020694018851772, 8.220504410870368472430258629997, 9.453516339000558347541420761782, 9.795200143776374715827789259878, 11.04275385952691672397487204842, 12.32276095697508825323849595748

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