Properties

Label 2-273-21.5-c1-0-13
Degree $2$
Conductor $273$
Sign $0.477 + 0.878i$
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  + (−2.13 − 1.23i)2-s + (1.47 + 0.901i)3-s + (2.04 + 3.54i)4-s + (1.98 − 3.43i)5-s + (−2.04 − 3.75i)6-s + (2.62 − 0.328i)7-s − 5.17i·8-s + (1.37 + 2.66i)9-s + (−8.46 + 4.88i)10-s + (−2.50 + 1.44i)11-s + (−0.170 + 7.09i)12-s i·13-s + (−6.01 − 2.53i)14-s + (6.02 − 3.28i)15-s + (−2.28 + 3.96i)16-s + (2.85 + 4.94i)17-s + ⋯
L(s)  = 1  + (−1.51 − 0.872i)2-s + (0.853 + 0.520i)3-s + (1.02 + 1.77i)4-s + (0.885 − 1.53i)5-s + (−0.836 − 1.53i)6-s + (0.992 − 0.124i)7-s − 1.82i·8-s + (0.457 + 0.889i)9-s + (−2.67 + 1.54i)10-s + (−0.754 + 0.435i)11-s + (−0.0490 + 2.04i)12-s − 0.277i·13-s + (−1.60 − 0.678i)14-s + (1.55 − 0.848i)15-s + (−0.572 + 0.991i)16-s + (0.692 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869158 - 0.517041i\)
\(L(\frac12)\) \(\approx\) \(0.869158 - 0.517041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 - 0.901i)T \)
7 \( 1 + (-2.62 + 0.328i)T \)
13 \( 1 + iT \)
good2 \( 1 + (2.13 + 1.23i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.98 + 3.43i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.50 - 1.44i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.85 - 4.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.00 + 1.15i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.664 + 0.383i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.11iT - 29T^{2} \)
31 \( 1 + (-0.564 + 0.326i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.28 + 2.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.44T + 41T^{2} \)
43 \( 1 + 8.81T + 43T^{2} \)
47 \( 1 + (1.84 - 3.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.17 + 2.98i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.198 + 0.343i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.42 - 1.97i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.12 - 1.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.63iT - 71T^{2} \)
73 \( 1 + (13.0 - 7.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.76 - 4.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 + (-3.59 + 6.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.08iT - 97T^{2} \)
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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

−11.44635156649361429769187775141, −10.19497821215376410084426155389, −10.00124883381183126647323215013, −8.820583562356778800620245639333, −8.361251506608670223758606807114, −7.69183421841346647177251864315, −5.41593739802820149189927139737, −4.24369513123788133483605600429, −2.36299904297735565662814494525, −1.43934396782571110045283178500, 1.72810737299691503285058903488, 2.90314700767558762634225365476, 5.55676992505910811200273899154, 6.63987521587885570357290169849, 7.35108373603708598776008176873, 8.087703443102198221989062117893, 9.047111110793480857006705345535, 9.987049047314646338440345777340, 10.63161790708496582618734883805, 11.66232889372837599532452695836

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