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Properties

Label 2-273-91.47-c1-0-17
Degree $2$
Conductor $273$
Sign $0.491 + 0.870i$
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  + (1.93 − 0.517i)2-s + (0.866 − 0.5i)3-s + (1.73 − 0.999i)4-s + (−0.960 − 3.58i)5-s + (1.41 − 1.41i)6-s + (−0.534 + 2.59i)7-s + (−0.000695 + 0.000695i)8-s + (0.499 − 0.866i)9-s + (−3.71 − 6.42i)10-s + (4.17 + 1.11i)11-s + (0.999 − 1.73i)12-s + (−3.27 − 1.51i)13-s + (0.309 + 5.28i)14-s + (−2.62 − 2.62i)15-s + (−2.00 + 3.46i)16-s + (3.45 + 5.98i)17-s + ⋯
L(s)  = 1  + (1.36 − 0.366i)2-s + (0.499 − 0.288i)3-s + (0.865 − 0.499i)4-s + (−0.429 − 1.60i)5-s + (0.577 − 0.577i)6-s + (−0.201 + 0.979i)7-s + (−0.000246 + 0.000246i)8-s + (0.166 − 0.288i)9-s + (−1.17 − 2.03i)10-s + (1.25 + 0.336i)11-s + (0.288 − 0.499i)12-s + (−0.907 − 0.419i)13-s + (0.0827 + 1.41i)14-s + (−0.677 − 0.677i)15-s + (−0.500 + 0.866i)16-s + (0.838 + 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.491 + 0.870i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.491 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25881 - 1.31896i\)
\(L(\frac12)\) \(\approx\) \(2.25881 - 1.31896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.534 - 2.59i)T \)
13 \( 1 + (3.27 + 1.51i)T \)
good2 \( 1 + (-1.93 + 0.517i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.960 + 3.58i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-4.17 - 1.11i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.45 - 5.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.927 + 3.46i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.10 - 0.635i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 + (-5.58 - 1.49i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.301 - 1.12i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.44 + 1.44i)T - 41iT^{2} \)
43 \( 1 + 2.89iT - 43T^{2} \)
47 \( 1 + (10.8 - 2.91i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.492 - 0.852i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.47 + 12.9i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (11.5 + 6.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.841 + 3.14i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.126 - 0.126i)T + 71iT^{2} \)
73 \( 1 + (1.61 - 6.02i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.82 - 3.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.62 - 2.62i)T - 83iT^{2} \)
89 \( 1 + (-10.4 + 2.79i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.18 + 3.18i)T - 97iT^{2} \)
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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

−12.29530272718912852870867552380, −11.45452054375505297594622460288, −9.634198432308947141110133132334, −8.833809884041338771465148286078, −8.043309020065450006988859002992, −6.40051664121807867740756949939, −5.30474018610842880087541824160, −4.45607916166094259279850873153, −3.36393192510073682029621033427, −1.78988725959084518239405512975, 2.92011849839551696648517109884, 3.65690243292850650080138775193, 4.55381477448931790862202237716, 6.16318759607815025336263208478, 7.03162621582010843711423363711, 7.57574802044848579302138256034, 9.453310994973209002115920698603, 10.20884578007508802465892636860, 11.44645065842163211398616121777, 11.99851589479424082111499595806

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