Properties

Label 2-273-91.5-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.983 + 0.181i$
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  + (0.703 + 2.62i)2-s + (0.866 − 0.5i)3-s + (−4.65 + 2.69i)4-s + (−1.03 + 0.276i)5-s + (1.92 + 1.92i)6-s + (−1.53 + 2.15i)7-s + (−6.49 − 6.49i)8-s + (0.499 − 0.866i)9-s + (−1.45 − 2.51i)10-s + (−1.12 + 4.19i)11-s + (−2.69 + 4.65i)12-s + (2.90 − 2.13i)13-s + (−6.73 − 2.51i)14-s + (−0.756 + 0.756i)15-s + (7.09 − 12.2i)16-s + (2.51 + 4.35i)17-s + ⋯
L(s)  = 1  + (0.497 + 1.85i)2-s + (0.499 − 0.288i)3-s + (−2.32 + 1.34i)4-s + (−0.462 + 0.123i)5-s + (0.784 + 0.784i)6-s + (−0.581 + 0.813i)7-s + (−2.29 − 2.29i)8-s + (0.166 − 0.288i)9-s + (−0.459 − 0.796i)10-s + (−0.338 + 1.26i)11-s + (−0.776 + 1.34i)12-s + (0.804 − 0.593i)13-s + (−1.79 − 0.673i)14-s + (−0.195 + 0.195i)15-s + (1.77 − 3.07i)16-s + (0.610 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.983 + 0.181i$
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: \((2,\ 273,\ (\ :1/2),\ -0.983 + 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121827 - 1.33061i\)
\(L(\frac12)\) \(\approx\) \(0.121827 - 1.33061i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 - 2.15i)T \)
13 \( 1 + (-2.90 + 2.13i)T \)
good2 \( 1 + (-0.703 - 2.62i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (1.03 - 0.276i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.12 - 4.19i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.51 - 4.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.21 + 0.861i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.25 + 1.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.78T + 29T^{2} \)
31 \( 1 + (0.156 - 0.585i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-6.15 + 1.64i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.51 - 2.51i)T + 41iT^{2} \)
43 \( 1 + 5.67iT - 43T^{2} \)
47 \( 1 + (-2.04 - 7.61i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.28 + 2.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.531 - 0.142i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.16 + 3.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.69 + 1.79i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.29 - 6.29i)T - 71iT^{2} \)
73 \( 1 + (4.24 + 1.13i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.13 + 7.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.35 - 6.35i)T + 83iT^{2} \)
89 \( 1 + (0.934 + 3.48i)T + (-77.0 + 44.5i)T^{2} \)
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

−12.68720085945581377185604213405, −12.11816331223088628522020959326, −10.05893396630078702343029466186, −9.090230669878907895541805223734, −8.118622398202459461213765589804, −7.57572204671230915307828932391, −6.45342339411019704243007312955, −5.66780651654540768520300381157, −4.35827560749139091603612324222, −3.18128489906041254406928008143, 0.905397891461142544491947680708, 2.90227600640786071639675717206, 3.63609169755322593638313426026, 4.56545139912317445102141504611, 5.95874214230826970849298098012, 7.896633687320996791949906249513, 8.957446046913946007140830443960, 9.828527162884849701740619214090, 10.56264228440509541321729567787, 11.47541190367462493017384609081

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