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Properties

Label 2-273-91.59-c1-0-8
Degree $2$
Conductor $273$
Sign $-0.110 - 0.993i$
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  + (1.85 + 1.85i)2-s + (0.866 − 0.5i)3-s + 4.91i·4-s + (0.502 + 0.134i)5-s + (2.53 + 0.680i)6-s + (−1.89 − 1.84i)7-s + (−5.41 + 5.41i)8-s + (0.499 − 0.866i)9-s + (0.684 + 1.18i)10-s + (−1.93 − 0.517i)11-s + (2.45 + 4.25i)12-s + (2.40 + 2.68i)13-s + (−0.0921 − 6.95i)14-s + (0.502 − 0.134i)15-s − 10.3·16-s + 5.09·17-s + ⋯
L(s)  = 1  + (1.31 + 1.31i)2-s + (0.499 − 0.288i)3-s + 2.45i·4-s + (0.224 + 0.0602i)5-s + (1.03 + 0.277i)6-s + (−0.716 − 0.697i)7-s + (−1.91 + 1.91i)8-s + (0.166 − 0.288i)9-s + (0.216 + 0.374i)10-s + (−0.582 − 0.156i)11-s + (0.709 + 1.22i)12-s + (0.667 + 0.744i)13-s + (−0.0246 − 1.85i)14-s + (0.129 − 0.0347i)15-s − 2.57·16-s + 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.110 - 0.993i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.110 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76119 + 1.96876i\)
\(L(\frac12)\) \(\approx\) \(1.76119 + 1.96876i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 + 1.84i)T \)
13 \( 1 + (-2.40 - 2.68i)T \)
good2 \( 1 + (-1.85 - 1.85i)T + 2iT^{2} \)
5 \( 1 + (-0.502 - 0.134i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.93 + 0.517i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 5.09T + 17T^{2} \)
19 \( 1 + (1.86 + 6.96i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 4.78iT - 23T^{2} \)
29 \( 1 + (1.35 - 2.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.867 - 3.23i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (5.25 - 5.25i)T - 37iT^{2} \)
41 \( 1 + (-0.938 - 3.50i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.62 - 2.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0215 + 0.0803i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.85 + 11.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.91 + 3.91i)T + 59iT^{2} \)
61 \( 1 + (0.652 + 0.376i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.81 - 10.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.33 - 5.00i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.05 + 0.551i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.17 - 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.09 + 7.09i)T - 83iT^{2} \)
89 \( 1 + (9.64 + 9.64i)T + 89iT^{2} \)
97 \( 1 + (3.28 + 0.880i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.74485551592370900194132972208, −11.64141264876491979224224495487, −10.21086216823439901329881835299, −8.842943775132587375975980048361, −7.947526223021500531702630479489, −6.88673165287671898961379998529, −6.40085617293384239038192517744, −5.08793620619979569467294625834, −3.92443388983074872860193357384, −2.90080146957259594864366034840, 1.87847759347150946885284929564, 3.17328488885341760071016080032, 3.85747761596617113510230258920, 5.54239541314921387225821652987, 5.82991547246538118793507007726, 7.84497591658453515952439515624, 9.303050374738053039367055153789, 10.06592534349962668920262965189, 10.69882510536668582962052064434, 12.00642904156962593914008830341

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