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Properties

Label 2-273-91.45-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.931 + 0.362i$
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  + (−0.783 + 0.783i)2-s + (−0.866 + 0.5i)3-s + 0.770i·4-s + (−0.994 + 3.71i)5-s + (0.286 − 1.07i)6-s + (0.0389 + 2.64i)7-s + (−2.17 − 2.17i)8-s + (0.499 − 0.866i)9-s + (−2.13 − 3.69i)10-s + (1.48 − 5.55i)11-s + (−0.385 − 0.667i)12-s + (2.56 + 2.53i)13-s + (−2.10 − 2.04i)14-s + (−0.994 − 3.71i)15-s + 1.86·16-s − 4.62·17-s + ⋯
L(s)  = 1  + (−0.554 + 0.554i)2-s + (−0.499 + 0.288i)3-s + 0.385i·4-s + (−0.444 + 1.66i)5-s + (0.117 − 0.437i)6-s + (0.0147 + 0.999i)7-s + (−0.767 − 0.767i)8-s + (0.166 − 0.288i)9-s + (−0.673 − 1.16i)10-s + (0.448 − 1.67i)11-s + (−0.111 − 0.192i)12-s + (0.711 + 0.702i)13-s + (−0.562 − 0.546i)14-s + (−0.256 − 0.958i)15-s + 0.465·16-s − 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.931 + 0.362i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.931 + 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107212 - 0.570856i\)
\(L(\frac12)\) \(\approx\) \(0.107212 - 0.570856i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0389 - 2.64i)T \)
13 \( 1 + (-2.56 - 2.53i)T \)
good2 \( 1 + (0.783 - 0.783i)T - 2iT^{2} \)
5 \( 1 + (0.994 - 3.71i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.48 + 5.55i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 4.62T + 17T^{2} \)
19 \( 1 + (-1.50 + 0.402i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 4.31iT - 23T^{2} \)
29 \( 1 + (2.38 - 4.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.70 + 0.993i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.78 + 1.78i)T + 37iT^{2} \)
41 \( 1 + (-4.40 + 1.18i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.65 - 1.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.89 - 2.11i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.36 - 2.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.95 - 9.95i)T - 59iT^{2} \)
61 \( 1 + (3.07 + 1.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.98 - 1.60i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.06 + 1.89i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.298 - 1.11i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.33 - 7.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.07 + 1.07i)T + 83iT^{2} \)
89 \( 1 + (-6.41 + 6.41i)T - 89iT^{2} \)
97 \( 1 + (1.63 - 6.11i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.98386388072283725876306692199, −11.35980266093530142856727923652, −10.80877695855246543301173354274, −9.292369365643745149334618664757, −8.666114917472880066421966119919, −7.47213238813130309861760482150, −6.46460418843609648395101404726, −5.96368251209633081201780464333, −3.83642009559503590722495665146, −2.94701552929161817769306739164, 0.58307747588731655359884362410, 1.72662411788498149021316483713, 4.26909924259656489945314790636, 4.98149592088895725379883861747, 6.36759433805846423023772049232, 7.65566191704257584661446733425, 8.650043084430999034602513725487, 9.567792634646356545516282442779, 10.41398006430644561989916126111, 11.36692690134610343241027249430

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