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Properties

Label 2-273-91.45-c1-0-11
Degree $2$
Conductor $273$
Sign $0.375 + 0.926i$
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  + (−1.03 + 1.03i)2-s + (−0.866 + 0.5i)3-s − 0.132i·4-s + (0.456 − 1.70i)5-s + (0.377 − 1.41i)6-s + (−2.58 + 0.554i)7-s + (−1.92 − 1.92i)8-s + (0.499 − 0.866i)9-s + (1.28 + 2.23i)10-s + (0.583 − 2.17i)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 + 4.24·16-s + 1.62·17-s + ⋯
L(s)  = 1  + (−0.730 + 0.730i)2-s + (−0.499 + 0.288i)3-s − 0.0663i·4-s + (0.204 − 0.762i)5-s + (0.154 − 0.575i)6-s + (−0.977 + 0.209i)7-s + (−0.681 − 0.681i)8-s + (0.166 − 0.288i)9-s + (0.407 + 0.705i)10-s + (0.176 − 0.656i)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 + 1.06·16-s + 0.394·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.375 + 0.926i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.375 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.277655 - 0.187088i\)
\(L(\frac12)\) \(\approx\) \(0.277655 - 0.187088i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 - 0.554i)T \)
13 \( 1 + (2.06 + 2.95i)T \)
good2 \( 1 + (1.03 - 1.03i)T - 2iT^{2} \)
5 \( 1 + (-0.456 + 1.70i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.583 + 2.17i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 1.62T + 17T^{2} \)
19 \( 1 + (6.94 - 1.85i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 6.50iT - 23T^{2} \)
29 \( 1 + (-3.90 + 6.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.30 + 1.68i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.48 + 1.48i)T + 37iT^{2} \)
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 \( 1 + (12.4 + 3.34i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.00 - 5.21i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.53 - 5.53i)T - 59iT^{2} \)
61 \( 1 + (5.42 + 3.13i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.12 - 1.37i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-6.88 - 1.84i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.22 + 4.55i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.67 + 8.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.82 + 3.82i)T + 83iT^{2} \)
89 \( 1 + (-3.59 + 3.59i)T - 89iT^{2} \)
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

−11.91059216892737807939413389955, −10.34520774213042781404603360200, −9.779512501989417430099929241629, −8.671281801948274996220291876367, −8.121186434714633473141902333863, −6.55430224221884006207874032026, −6.08451566719863766919468698015, −4.65806932025633925888005551654, −3.13540167823597467636609680892, −0.33217439138111836607153085175, 1.81775106283018905381129721501, 3.15000864947123025350341611981, 4.95486854942007518833291682339, 6.49928619870822094219398738625, 6.86354657488421278233123492627, 8.480973118577396698993871044341, 9.664155700682571567070029650607, 10.14783057467645527501383901003, 10.97384002274937463645109114097, 11.91274829678091249475433315234

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