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Properties

Label 2-273-91.5-c1-0-5
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: $\chi_{273} (187, \cdot )$
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

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

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