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Properties

Label 2-273-91.6-c1-0-16
Degree $2$
Conductor $273$
Sign $-0.979 + 0.201i$
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  + (−0.189 − 0.707i)2-s + (−0.866 − 0.5i)3-s + (1.26 − 0.731i)4-s + (−1.23 − 1.23i)5-s + (−0.189 + 0.707i)6-s + (−2.64 − 0.0736i)7-s + (−1.79 − 1.79i)8-s + (0.499 + 0.866i)9-s + (−0.639 + 1.10i)10-s + (−3.18 + 0.854i)11-s − 1.46·12-s + (2.53 − 2.56i)13-s + (0.449 + 1.88i)14-s + (0.451 + 1.68i)15-s + (0.532 − 0.923i)16-s + (−0.433 − 0.751i)17-s + ⋯
L(s)  = 1  + (−0.134 − 0.500i)2-s + (−0.499 − 0.288i)3-s + (0.633 − 0.365i)4-s + (−0.551 − 0.551i)5-s + (−0.0774 + 0.289i)6-s + (−0.999 − 0.0278i)7-s + (−0.634 − 0.634i)8-s + (0.166 + 0.288i)9-s + (−0.202 + 0.350i)10-s + (−0.961 + 0.257i)11-s − 0.422·12-s + (0.701 − 0.712i)13-s + (0.120 + 0.504i)14-s + (0.116 + 0.434i)15-s + (0.133 − 0.230i)16-s + (−0.105 − 0.182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.979 + 0.201i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.979 + 0.201i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0691142 - 0.679043i\)
\(L(\frac12)\) \(\approx\) \(0.0691142 - 0.679043i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.0736i)T \)
13 \( 1 + (-2.53 + 2.56i)T \)
good2 \( 1 + (0.189 + 0.707i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (1.23 + 1.23i)T + 5iT^{2} \)
11 \( 1 + (3.18 - 0.854i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.433 + 0.751i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.01 - 3.80i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.77 + 2.17i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.65 + 4.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.220 - 0.220i)T + 31iT^{2} \)
37 \( 1 + (3.57 - 0.957i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.90 + 0.509i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9.99 + 5.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.68 + 3.68i)T - 47iT^{2} \)
53 \( 1 + 3.55T + 53T^{2} \)
59 \( 1 + (-8.89 - 2.38i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.78 + 2.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.01 - 3.80i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-11.9 - 3.20i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.55 + 5.55i)T - 73iT^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + (3.80 + 3.80i)T + 83iT^{2} \)
89 \( 1 + (3.26 + 12.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.756 + 2.82i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.52003306989356351058515479773, −10.41887976601276814591528074549, −10.03903703772467334543400168187, −8.554303687038004334951407696080, −7.50912493789649040878300352980, −6.34858303007180539876919575237, −5.55655988576625842993320160066, −3.90934663825557238581525933841, −2.45954751534529672509280629412, −0.54396515224327914999066483903, 2.76479648826756850257972708903, 3.87126201858143572265555919357, 5.59443924456019959448531807576, 6.54446163006307219419588566143, 7.23266977408775697374334470804, 8.353247076072949812058504201949, 9.464582947166849671617905856378, 10.78323464612966749880915893766, 11.18604127399694867060276369873, 12.24546981754649816673902951980

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