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Properties

Label 2-273-91.34-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.193 - 0.981i$
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  + (−1.94 − 1.94i)2-s + i·3-s + 5.57i·4-s + (0.136 − 0.136i)5-s + (1.94 − 1.94i)6-s + (−2.24 − 1.39i)7-s + (6.96 − 6.96i)8-s − 9-s − 0.532·10-s + (0.555 − 0.555i)11-s − 5.57·12-s + (−1.81 + 3.11i)13-s + (1.65 + 7.09i)14-s + (0.136 + 0.136i)15-s − 15.9·16-s − 4.49·17-s + ⋯
L(s)  = 1  + (−1.37 − 1.37i)2-s + 0.577i·3-s + 2.78i·4-s + (0.0611 − 0.0611i)5-s + (0.794 − 0.794i)6-s + (−0.849 − 0.527i)7-s + (2.46 − 2.46i)8-s − 0.333·9-s − 0.168·10-s + (0.167 − 0.167i)11-s − 1.61·12-s + (−0.502 + 0.864i)13-s + (0.442 + 1.89i)14-s + (0.0353 + 0.0353i)15-s − 3.98·16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.193 - 0.981i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.193 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0901265 + 0.109639i\)
\(L(\frac12)\) \(\approx\) \(0.0901265 + 0.109639i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.24 + 1.39i)T \)
13 \( 1 + (1.81 - 3.11i)T \)
good2 \( 1 + (1.94 + 1.94i)T + 2iT^{2} \)
5 \( 1 + (-0.136 + 0.136i)T - 5iT^{2} \)
11 \( 1 + (-0.555 + 0.555i)T - 11iT^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
19 \( 1 + (4.15 - 4.15i)T - 19iT^{2} \)
23 \( 1 + 0.423iT - 23T^{2} \)
29 \( 1 + 7.01T + 29T^{2} \)
31 \( 1 + (-0.273 + 0.273i)T - 31iT^{2} \)
37 \( 1 + (-5.75 + 5.75i)T - 37iT^{2} \)
41 \( 1 + (7.29 - 7.29i)T - 41iT^{2} \)
43 \( 1 + 1.86iT - 43T^{2} \)
47 \( 1 + (4.75 + 4.75i)T + 47iT^{2} \)
53 \( 1 + 4.31T + 53T^{2} \)
59 \( 1 + (0.691 + 0.691i)T + 59iT^{2} \)
61 \( 1 - 8.11iT - 61T^{2} \)
67 \( 1 + (-0.837 - 0.837i)T + 67iT^{2} \)
71 \( 1 + (-1.00 - 1.00i)T + 71iT^{2} \)
73 \( 1 + (6.81 + 6.81i)T + 73iT^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + (-6.91 + 6.91i)T - 83iT^{2} \)
89 \( 1 + (-6.46 - 6.46i)T + 89iT^{2} \)
97 \( 1 + (-3.19 + 3.19i)T - 97iT^{2} \)
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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

−11.75220354847352778503181806139, −11.01464429921534460272017490909, −10.22791861972816939052471010408, −9.430464743254072790540247633181, −8.894634886634909896547058353903, −7.68074547463938035873804658266, −6.58089098210924277055194784308, −4.30671900577732724651621480903, −3.45085175815979316852245023718, −1.99939164197651085363288669054, 0.15816066960336783384088723287, 2.27143444553231705896842230549, 4.99051455162277512579312229747, 6.22218154480950285709613497052, 6.69749326421374625901350505163, 7.74541053712187460423160847079, 8.666159441360149646095871184615, 9.393311014378645775165645373485, 10.30601053830147363145917184091, 11.30009619261748334652128749241

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