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Properties

Label 2-273-273.44-c1-0-14
Degree $2$
Conductor $273$
Sign $0.322 - 0.946i$
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  + (−0.965 + 0.258i)2-s + (0.599 + 1.62i)3-s + (−0.866 + 0.5i)4-s + (3.29 − 0.883i)5-s + (−1 − 1.41i)6-s + (2.38 + 1.15i)7-s + (2.12 − 2.12i)8-s + (−2.28 + 1.94i)9-s + (−2.95 + 1.70i)10-s + (0.165 + 0.0444i)11-s + (−1.33 − 1.10i)12-s + (2 − 3i)13-s + (−2.59 − 0.500i)14-s + (3.41 + 4.82i)15-s + (−0.500 + 0.866i)16-s + (−1.91 − 3.31i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.346 + 0.938i)3-s + (−0.433 + 0.250i)4-s + (1.47 − 0.395i)5-s + (−0.408 − 0.577i)6-s + (0.899 + 0.436i)7-s + (0.749 − 0.749i)8-s + (−0.760 + 0.649i)9-s + (−0.935 + 0.539i)10-s + (0.0499 + 0.0133i)11-s + (−0.384 − 0.319i)12-s + (0.554 − 0.832i)13-s + (−0.694 − 0.133i)14-s + (0.881 + 1.24i)15-s + (−0.125 + 0.216i)16-s + (−0.464 − 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.322 - 0.946i$
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: \((2,\ 273,\ (\ :1/2),\ 0.322 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970722 + 0.694419i\)
\(L(\frac12)\) \(\approx\) \(0.970722 + 0.694419i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.599 - 1.62i)T \)
7 \( 1 + (-2.38 - 1.15i)T \)
13 \( 1 + (-2 + 3i)T \)
good2 \( 1 + (0.965 - 0.258i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-3.29 + 0.883i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.165 - 0.0444i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.91 + 3.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.20 - 4.49i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.70 - 8.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - iT - 29T^{2} \)
31 \( 1 + (4.66 + 1.24i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.86 + 1.03i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (4 + 4i)T + 41iT^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + (2.75 + 10.2i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.30 - 1.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.23 - 1.93i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.36 + 0.902i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.53 + 7.53i)T + 71iT^{2} \)
73 \( 1 + (-1.85 + 6.92i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.707 + 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.0 - 11.0i)T + 83iT^{2} \)
89 \( 1 + (1.16 + 4.33i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−11.98465190027027523468758344764, −10.71292321896568222216608235676, −9.889499106982270900791179684948, −9.270193078990427449643900092768, −8.533444975322923058066969355963, −7.66013645637483668153504939886, −5.70951502259515345541959361802, −5.12864017869758130985489802473, −3.69200176251436938062739059543, −1.85972252475738789084191049590, 1.39957010743913435324796569742, 2.29377289293568670451295488936, 4.51823481458443080902897497854, 5.93776730686529426592563716479, 6.77683823136719023778053116492, 8.104323698982476849660858600019, 8.832363842141743387637469103671, 9.698454006842534124398033418392, 10.70434481277516550323661118890, 11.41734541789303205151032257528

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