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Properties

Label 2-273-91.73-c1-0-14
Degree $2$
Conductor $273$
Sign $-0.665 + 0.746i$
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  + (0.549 − 2.05i)2-s + (−0.866 − 0.5i)3-s + (−2.17 − 1.25i)4-s + (3.81 + 1.02i)5-s + (−1.50 + 1.50i)6-s + (−0.231 − 2.63i)7-s + (−0.766 + 0.766i)8-s + (0.499 + 0.866i)9-s + (4.19 − 7.25i)10-s + (−0.403 − 1.50i)11-s + (1.25 + 2.17i)12-s + (−2.17 + 2.87i)13-s + (−5.53 − 0.973i)14-s + (−2.79 − 2.79i)15-s + (−1.35 − 2.35i)16-s + (−1.00 + 1.73i)17-s + ⋯
L(s)  = 1  + (0.388 − 1.45i)2-s + (−0.499 − 0.288i)3-s + (−1.08 − 0.627i)4-s + (1.70 + 0.456i)5-s + (−0.613 + 0.613i)6-s + (−0.0875 − 0.996i)7-s + (−0.270 + 0.270i)8-s + (0.166 + 0.288i)9-s + (1.32 − 2.29i)10-s + (−0.121 − 0.454i)11-s + (0.362 + 0.627i)12-s + (−0.603 + 0.797i)13-s + (−1.47 − 0.260i)14-s + (−0.720 − 0.720i)15-s + (−0.339 − 0.588i)16-s + (−0.243 + 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.665 + 0.746i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.665 + 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663978 - 1.48085i\)
\(L(\frac12)\) \(\approx\) \(0.663978 - 1.48085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.231 + 2.63i)T \)
13 \( 1 + (2.17 - 2.87i)T \)
good2 \( 1 + (-0.549 + 2.05i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-3.81 - 1.02i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.403 + 1.50i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.00 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.00 - 0.806i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.40 - 1.96i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.87T + 29T^{2} \)
31 \( 1 + (-0.375 - 1.40i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-7.76 - 2.08i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.75 + 6.75i)T - 41iT^{2} \)
43 \( 1 - 3.27iT - 43T^{2} \)
47 \( 1 + (0.00206 - 0.00770i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.83 - 11.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.32 + 2.49i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.86 + 4.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.31 + 1.15i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.217 + 0.217i)T + 71iT^{2} \)
73 \( 1 + (-4.92 + 1.31i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.44 + 9.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.4 - 11.4i)T - 83iT^{2} \)
89 \( 1 + (2.74 - 10.2i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.68 - 5.68i)T - 97iT^{2} \)
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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

−11.33731852794228534702719566451, −10.80412536870659389969955912585, −9.884345522964242094174149138691, −9.451146531541246876862125861717, −7.42947462911814661271276739266, −6.36504597124305521958497299273, −5.30689704526617841721743152710, −3.96923087198033423759165440447, −2.50130840044691235160036183590, −1.41556018603145807359658415369, 2.31804589249773193429055439196, 4.69928639910255464872289680242, 5.55912379911031262862026533384, 5.88397875383943906241072486986, 7.02577742027959625933714243500, 8.293534521454012307684071060572, 9.433334214664006067420612968179, 9.905374685438699750036884773857, 11.37085077550292567799304618735, 12.78629651670002117492826795693

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