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Properties

Label 2-273-91.80-c1-0-8
Degree $2$
Conductor $273$
Sign $0.995 - 0.0918i$
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  + (0.0407 + 0.151i)2-s − i·3-s + (1.71 − 0.987i)4-s + (−0.570 + 2.13i)5-s + (0.151 − 0.0407i)6-s + (0.961 + 2.46i)7-s + (0.442 + 0.442i)8-s − 9-s − 0.346·10-s + (3.97 + 3.97i)11-s + (−0.987 − 1.71i)12-s + (1.34 − 3.34i)13-s + (−0.335 + 0.246i)14-s + (2.13 + 0.570i)15-s + (1.92 − 3.33i)16-s + (−1.79 − 3.11i)17-s + ⋯
L(s)  = 1  + (0.0287 + 0.107i)2-s − 0.577i·3-s + (0.855 − 0.493i)4-s + (−0.255 + 0.952i)5-s + (0.0620 − 0.0166i)6-s + (0.363 + 0.931i)7-s + (0.156 + 0.156i)8-s − 0.333·9-s − 0.109·10-s + (1.19 + 1.19i)11-s + (−0.285 − 0.493i)12-s + (0.371 − 0.928i)13-s + (−0.0896 + 0.0658i)14-s + (0.550 + 0.147i)15-s + (0.481 − 0.834i)16-s + (−0.435 − 0.754i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.995 - 0.0918i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.995 - 0.0918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56510 + 0.0719932i\)
\(L(\frac12)\) \(\approx\) \(1.56510 + 0.0719932i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (-0.961 - 2.46i)T \)
13 \( 1 + (-1.34 + 3.34i)T \)
good2 \( 1 + (-0.0407 - 0.151i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.570 - 2.13i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.97 - 3.97i)T + 11iT^{2} \)
17 \( 1 + (1.79 + 3.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.40 + 3.40i)T + 19iT^{2} \)
23 \( 1 + (3.85 + 2.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.02 - 1.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.40 + 0.643i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (9.86 - 2.64i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.34 - 5.01i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.12 - 2.38i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.56 + 1.22i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.201 + 0.348i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.64 + 0.707i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 1.86iT - 61T^{2} \)
67 \( 1 + (-4.87 + 4.87i)T - 67iT^{2} \)
71 \( 1 + (3.63 + 13.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.19 + 4.45i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.45 + 4.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.3 + 10.3i)T + 83iT^{2} \)
89 \( 1 + (-4.67 - 17.4i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.52 + 0.944i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.83945788090869608482358827292, −11.13907663917725667849541052470, −10.22514922205553839701883412116, −9.003090115839857628306194600991, −7.78669419524265365219393288161, −6.77136445483084775571888180722, −6.32267179622268364700543237850, −4.90004921853130578058685373079, −2.95560603844445831582950453870, −1.87674693977500828508658098235, 1.55740080924214911697717307924, 3.77856466603882728541004397182, 4.17024244308899939332881281742, 5.93291183109250146547566850528, 6.91098657930563248988870279071, 8.342477161788811680170914095809, 8.715410406477365150241291777990, 10.19656580635677761908972793911, 11.09856109266177489783412145095, 11.72868883911421549341211874289

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