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Properties

Label 2-273-91.51-c1-0-8
Degree $2$
Conductor $273$
Sign $0.988 - 0.152i$
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  + (1.14 + 0.661i)2-s + (−0.5 − 0.866i)3-s + (−0.124 − 0.214i)4-s + (1.98 + 1.14i)5-s − 1.32i·6-s + (1.14 + 2.38i)7-s − 2.97i·8-s + (−0.499 + 0.866i)9-s + (1.51 + 2.62i)10-s + (1.08 − 0.626i)11-s + (−0.124 + 0.214i)12-s + (2.78 − 2.29i)13-s + (−0.264 + 3.49i)14-s − 2.29i·15-s + (1.72 − 2.98i)16-s + (−0.418 − 0.725i)17-s + ⋯
L(s)  = 1  + (0.810 + 0.467i)2-s + (−0.288 − 0.499i)3-s + (−0.0620 − 0.107i)4-s + (0.887 + 0.512i)5-s − 0.540i·6-s + (0.433 + 0.901i)7-s − 1.05i·8-s + (−0.166 + 0.288i)9-s + (0.479 + 0.830i)10-s + (0.327 − 0.188i)11-s + (−0.0357 + 0.0620i)12-s + (0.772 − 0.635i)13-s + (−0.0705 + 0.933i)14-s − 0.591i·15-s + (0.430 − 0.745i)16-s + (−0.101 − 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.988 - 0.152i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.988 - 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93389 + 0.148008i\)
\(L(\frac12)\) \(\approx\) \(1.93389 + 0.148008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.14 - 2.38i)T \)
13 \( 1 + (-2.78 + 2.29i)T \)
good2 \( 1 + (-1.14 - 0.661i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.98 - 1.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.08 + 0.626i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.418 + 0.725i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.837 - 0.483i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.11 - 7.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.78T + 29T^{2} \)
31 \( 1 + (0.553 - 0.319i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.21 + 3.01i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.497iT - 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 + (-7.83 - 4.52i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.04 + 5.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.97 - 4.60i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.36 + 5.82i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 5.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (-3.04 + 1.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.71 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.79iT - 83T^{2} \)
89 \( 1 + (-3.47 - 2.00i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.57iT - 97T^{2} \)
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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

−12.10363064894267436440547803213, −11.14423100271098592310136209104, −10.04226703051919712620001680136, −9.111250189183353818361783199263, −7.80529623159472533992937317582, −6.53845684397701005940843758017, −5.82076371984097976560692215818, −5.23722764820263736097150433469, −3.50427109701050013676640354159, −1.77611838853621486981409085334, 1.83576891528729060117202252886, 3.69759886034635679427666634633, 4.50609113008314477737886914681, 5.45319593371569757108888410842, 6.63909885236760681515583515537, 8.203524860651299919483421766387, 9.107038735436231575580840094463, 10.21075823636390367787329482900, 11.06433133358293734535739075466, 11.91723688531429335734878945979

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