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Properties

Label 2-273-91.76-c1-0-1
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: $\chi_{273} (76, \cdot )$
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

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

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