Properties

Label 2-273-273.44-c1-0-16
Degree $2$
Conductor $273$
Sign $0.950 + 0.311i$
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  + (−1.64 + 0.439i)2-s + (1.02 − 1.39i)3-s + (0.769 − 0.444i)4-s + (3.49 − 0.936i)5-s + (−1.06 + 2.74i)6-s + (0.612 + 2.57i)7-s + (1.33 − 1.33i)8-s + (−0.914 − 2.85i)9-s + (−5.32 + 3.07i)10-s + (0.378 + 0.101i)11-s + (0.164 − 1.53i)12-s + (−0.988 + 3.46i)13-s + (−2.13 − 3.95i)14-s + (2.25 − 5.84i)15-s + (−2.49 + 4.31i)16-s + (0.773 + 1.33i)17-s + ⋯
L(s)  = 1  + (−1.16 + 0.311i)2-s + (0.589 − 0.807i)3-s + (0.384 − 0.222i)4-s + (1.56 − 0.419i)5-s + (−0.433 + 1.12i)6-s + (0.231 + 0.972i)7-s + (0.472 − 0.472i)8-s + (−0.304 − 0.952i)9-s + (−1.68 + 0.972i)10-s + (0.114 + 0.0305i)11-s + (0.0474 − 0.441i)12-s + (−0.274 + 0.961i)13-s + (−0.571 − 1.05i)14-s + (0.583 − 1.51i)15-s + (−0.623 + 1.07i)16-s + (0.187 + 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.950 + 0.311i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.950 + 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05721 - 0.169117i\)
\(L(\frac12)\) \(\approx\) \(1.05721 - 0.169117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.02 + 1.39i)T \)
7 \( 1 + (-0.612 - 2.57i)T \)
13 \( 1 + (0.988 - 3.46i)T \)
good2 \( 1 + (1.64 - 0.439i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-3.49 + 0.936i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.378 - 0.101i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.773 - 1.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.50 + 5.61i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.82 + 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.29iT - 29T^{2} \)
31 \( 1 + (-7.37 - 1.97i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.893 - 0.239i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.40 + 5.40i)T + 41iT^{2} \)
43 \( 1 - 2.78iT - 43T^{2} \)
47 \( 1 + (1.72 + 6.45i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.185 + 0.107i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.75 + 2.07i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.29 - 9.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.139 + 0.0374i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.99 + 2.99i)T + 71iT^{2} \)
73 \( 1 + (-1.27 + 4.77i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.10 - 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.55 + 5.55i)T + 83iT^{2} \)
89 \( 1 + (-3.50 - 13.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.33 - 6.33i)T - 97iT^{2} \)
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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

−11.99464845256690331823914232964, −10.52587822719520813937272333546, −9.435079528734046254348073702499, −8.936400261496659174271612349455, −8.424471577578118154539674472146, −6.94201790923258772934784937542, −6.32630844190375566711130588082, −4.88273623318395876865639424258, −2.50504201686196513522882041011, −1.45945287185633888714225884322, 1.64111265993273657074452578171, 3.02105619003132744437228044351, 4.72688400549294856129550545670, 5.89594252652649202304921494325, 7.50435102075328196794007263346, 8.304521539316382893979654378543, 9.496417927576270635222443773034, 10.03370864510913242268710659073, 10.38848145315005307461370654785, 11.35310842244252482253980627392

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