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Properties

Label 2-273-91.45-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.867 + 0.497i$
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  + (−1.79 + 1.79i)2-s + (0.866 − 0.5i)3-s − 4.44i·4-s + (−0.590 + 2.20i)5-s + (−0.657 + 2.45i)6-s + (−2.51 − 0.826i)7-s + (4.38 + 4.38i)8-s + (0.499 − 0.866i)9-s + (−2.89 − 5.02i)10-s + (−0.449 + 1.67i)11-s + (−2.22 − 3.84i)12-s + (−1.05 + 3.44i)13-s + (5.99 − 3.02i)14-s + (0.590 + 2.20i)15-s − 6.86·16-s − 7.51·17-s + ⋯
L(s)  = 1  + (−1.26 + 1.26i)2-s + (0.499 − 0.288i)3-s − 2.22i·4-s + (−0.264 + 0.986i)5-s + (−0.268 + 1.00i)6-s + (−0.949 − 0.312i)7-s + (1.55 + 1.55i)8-s + (0.166 − 0.288i)9-s + (−0.916 − 1.58i)10-s + (−0.135 + 0.505i)11-s + (−0.641 − 1.11i)12-s + (−0.292 + 0.956i)13-s + (1.60 − 0.809i)14-s + (0.152 + 0.569i)15-s − 1.71·16-s − 1.82·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.867 + 0.497i$
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: \((2,\ 273,\ (\ :1/2),\ -0.867 + 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0744573 - 0.279228i\)
\(L(\frac12)\) \(\approx\) \(0.0744573 - 0.279228i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.51 + 0.826i)T \)
13 \( 1 + (1.05 - 3.44i)T \)
good2 \( 1 + (1.79 - 1.79i)T - 2iT^{2} \)
5 \( 1 + (0.590 - 2.20i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.449 - 1.67i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 7.51T + 17T^{2} \)
19 \( 1 + (5.80 - 1.55i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.0216iT - 23T^{2} \)
29 \( 1 + (-0.432 + 0.749i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.81 + 1.55i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.64 + 6.64i)T + 37iT^{2} \)
41 \( 1 + (2.45 - 0.656i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.71 + 0.987i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.62 - 1.50i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.87 - 11.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.10 - 4.10i)T - 59iT^{2} \)
61 \( 1 + (-3.12 - 1.80i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.08 - 0.290i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-12.5 - 3.36i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.56 - 9.58i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.16 - 3.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.64 + 8.64i)T + 83iT^{2} \)
89 \( 1 + (-1.71 + 1.71i)T - 89iT^{2} \)
97 \( 1 + (1.99 - 7.43i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.55056947136159974750945089103, −10.99229085635123304115851088121, −10.26995475499442085885474605142, −9.307846009988693651405779604378, −8.614707208705579813990056987481, −7.38976257574171538820074785494, −6.80520748883211759293087819592, −6.27452632905060626643591578306, −4.24588478987956586983420207706, −2.31208936790744204486925615665, 0.29256605251244013517154864423, 2.29914022644145651127250590273, 3.38991385757997689017090197535, 4.71283135145294561056274522096, 6.67341308267078568141977006946, 8.318768480341030075918191416573, 8.572145202200074070700494551720, 9.415717100970299780232619275332, 10.32444386121829223722759346433, 11.07559906669060949562933590329

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