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Properties

Label 2-273-91.59-c1-0-9
Degree $2$
Conductor $273$
Sign $0.923 + 0.383i$
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  + (0.430 + 0.430i)2-s + (−0.866 + 0.5i)3-s − 1.62i·4-s + (1.97 + 0.529i)5-s + (−0.588 − 0.157i)6-s + (−1.23 − 2.34i)7-s + (1.56 − 1.56i)8-s + (0.499 − 0.866i)9-s + (0.623 + 1.08i)10-s + (−0.0981 − 0.0263i)11-s + (0.814 + 1.41i)12-s + (3.09 − 1.85i)13-s + (0.476 − 1.53i)14-s + (−1.97 + 0.529i)15-s − 1.91·16-s + 3.63·17-s + ⋯
L(s)  = 1  + (0.304 + 0.304i)2-s + (−0.499 + 0.288i)3-s − 0.814i·4-s + (0.884 + 0.236i)5-s + (−0.240 − 0.0643i)6-s + (−0.466 − 0.884i)7-s + (0.552 − 0.552i)8-s + (0.166 − 0.288i)9-s + (0.197 + 0.341i)10-s + (−0.0295 − 0.00792i)11-s + (0.235 + 0.407i)12-s + (0.858 − 0.513i)13-s + (0.127 − 0.411i)14-s + (−0.510 + 0.136i)15-s − 0.477·16-s + 0.880·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41301 - 0.282068i\)
\(L(\frac12)\) \(\approx\) \(1.41301 - 0.282068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (1.23 + 2.34i)T \)
13 \( 1 + (-3.09 + 1.85i)T \)
good2 \( 1 + (-0.430 - 0.430i)T + 2iT^{2} \)
5 \( 1 + (-1.97 - 0.529i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0981 + 0.0263i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 3.63T + 17T^{2} \)
19 \( 1 + (-0.374 - 1.39i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 4.80iT - 23T^{2} \)
29 \( 1 + (-3.79 + 6.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.73 - 6.47i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.15 - 2.15i)T - 37iT^{2} \)
41 \( 1 + (0.872 + 3.25i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.21 - 4.16i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.529 - 1.97i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.27 - 9.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.58 + 1.58i)T + 59iT^{2} \)
61 \( 1 + (5.15 + 2.97i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.62 - 6.05i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.0733 - 0.273i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.17 + 1.11i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.01 + 8.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.74 + 3.74i)T - 83iT^{2} \)
89 \( 1 + (-5.75 - 5.75i)T + 89iT^{2} \)
97 \( 1 + (-16.5 - 4.43i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.74064191198185102482366741937, −10.46731213812830856371958746596, −10.25075236305753876877727985038, −9.368638973828398961245928306122, −7.71361512616280567301805936159, −6.43983898964869620495132874699, −5.94852287472239013394826228912, −4.87339144003498511634115760655, −3.48852918365893782214633164946, −1.26608060100203405912754613309, 1.96013504463297828469657491392, 3.29632467992896838401522816482, 4.85025446330234435205507675138, 5.90440755596886002471142971977, 6.82469510766671389770597914042, 8.223532330267115900161091596588, 9.054794702470740161564305603185, 10.14602485588775821673216265607, 11.31291588184423404699666347477, 12.07858720262176259854338993213

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