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Properties

Label 2-273-91.88-c1-0-5
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  + (1.83 + 1.05i)2-s − 3-s + (1.24 + 2.15i)4-s + (−3.01 + 1.74i)5-s + (−1.83 − 1.05i)6-s + (1.38 + 2.25i)7-s + 1.04i·8-s + 9-s − 7.39·10-s + 4.16i·11-s + (−1.24 − 2.15i)12-s + (1.86 + 3.08i)13-s + (0.164 + 5.60i)14-s + (3.01 − 1.74i)15-s + (1.38 − 2.39i)16-s + (−1.38 − 2.40i)17-s + ⋯
L(s)  = 1  + (1.29 + 0.749i)2-s − 0.577·3-s + (0.623 + 1.07i)4-s + (−1.35 + 0.779i)5-s + (−0.749 − 0.432i)6-s + (0.525 + 0.850i)7-s + 0.369i·8-s + 0.333·9-s − 2.33·10-s + 1.25i·11-s + (−0.359 − 0.623i)12-s + (0.518 + 0.855i)13-s + (0.0440 + 1.49i)14-s + (0.779 − 0.450i)15-s + (0.346 − 0.599i)16-s + (−0.337 − 0.583i)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: $\chi_{273} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.880134 + 1.48087i\)
\(L(\frac12)\) \(\approx\) \(0.880134 + 1.48087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (-1.38 - 2.25i)T \)
13 \( 1 + (-1.86 - 3.08i)T \)
good2 \( 1 + (-1.83 - 1.05i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (3.01 - 1.74i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 4.16iT - 11T^{2} \)
17 \( 1 + (1.38 + 2.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 5.64iT - 19T^{2} \)
23 \( 1 + (-1.83 + 3.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.07 + 1.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.58 - 4.38i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.28 - 1.89i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.80 + 2.77i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.45 - 4.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.46 - 0.843i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.45 - 5.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.67 - 5.58i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.32T + 61T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 + (-5.36 - 3.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.790 + 0.456i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.91 + 8.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.95iT - 83T^{2} \)
89 \( 1 + (0.00476 + 0.00275i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.9 - 7.47i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.07047341961284964191040043775, −11.71503687840451336403771812269, −10.80832912149206261537942137732, −9.244359641884211279523475554846, −7.82747423832655891921417535012, −6.96830481483811655113730006209, −6.33754268382992600902539698730, −4.74930357534477407288707331539, −4.44029980152801618783824095546, −2.84008153416308618624419440155, 1.05954715360608222701448977553, 3.50414197213039647078141559738, 4.09512008620770016292292664319, 5.13800549729820812496832854124, 6.11601980295405128835537954336, 7.85108953238933536049612677054, 8.377227707856801169276689573331, 10.31529896040027525326585540586, 11.24797441338949092984282673732, 11.48312551102510988790980397997

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