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Properties

Label 2-273-91.19-c1-0-8
Degree $2$
Conductor $273$
Sign $0.727 - 0.686i$
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  + (1.41 + 0.377i)2-s + i·3-s + (0.114 + 0.0663i)4-s + (1.70 − 0.456i)5-s + (−0.377 + 1.41i)6-s + (1.96 + 1.77i)7-s + (−1.92 − 1.92i)8-s − 9-s + 2.57·10-s + (1.59 + 1.59i)11-s + (−0.0663 + 0.114i)12-s + (2.06 + 2.95i)13-s + (2.09 + 3.24i)14-s + (0.456 + 1.70i)15-s + (−2.12 − 3.67i)16-s + (0.813 − 1.40i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.267i)2-s + 0.577i·3-s + (0.0574 + 0.0331i)4-s + (0.762 − 0.204i)5-s + (−0.154 + 0.575i)6-s + (0.742 + 0.670i)7-s + (−0.681 − 0.681i)8-s − 0.333·9-s + 0.815·10-s + (0.480 + 0.480i)11-s + (−0.0191 + 0.0331i)12-s + (0.572 + 0.819i)13-s + (0.561 + 0.866i)14-s + (0.117 + 0.440i)15-s + (−0.530 − 0.919i)16-s + (0.197 − 0.341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.727 - 0.686i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.727 - 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01248 + 0.799876i\)
\(L(\frac12)\) \(\approx\) \(2.01248 + 0.799876i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 - 1.77i)T \)
13 \( 1 + (-2.06 - 2.95i)T \)
good2 \( 1 + (-1.41 - 0.377i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.70 + 0.456i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.59 - 1.59i)T + 11iT^{2} \)
17 \( 1 + (-0.813 + 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.08 + 5.08i)T + 19iT^{2} \)
23 \( 1 + (5.63 - 3.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.90 + 6.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.68 + 6.30i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.545 - 2.03i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-6.84 + 1.83i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.48 - 5.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.34 + 12.4i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.00 + 5.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.02 - 7.55i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 6.26iT - 61T^{2} \)
67 \( 1 + (3.75 - 3.75i)T - 67iT^{2} \)
71 \( 1 + (-6.88 - 1.84i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.55 + 1.22i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.67 - 8.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.82 - 3.82i)T + 83iT^{2} \)
89 \( 1 + (-4.91 - 1.31i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.57 - 9.62i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.97177790318019175123278963651, −11.42107388111897079259802720024, −9.882284806248452935782574300374, −9.326492500370351336447530829689, −8.324077723996478115169406763520, −6.56964933323414021930977444771, −5.79447825313752893482448569899, −4.78929510675130959873385799576, −4.01971099025786782976056256468, −2.19013435316696880796005286829, 1.73149452101168173080332797301, 3.29839674769786452184709659806, 4.45240739960731379467223446620, 5.77264531695253261281278677261, 6.40412053043989985682078206974, 8.035525099086798231340679784532, 8.625306260530524252906004193805, 10.26801122211718816392883056659, 10.94630427548990893646203215967, 12.14130080202671993555734552590

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