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Properties

Label 2-273-273.44-c1-0-33
Degree $2$
Conductor $273$
Sign $-0.939 - 0.341i$
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  + (0.623 − 0.167i)2-s + (0.313 − 1.70i)3-s + (−1.37 + 0.791i)4-s + (−2.98 + 0.801i)5-s + (−0.0887 − 1.11i)6-s + (−2.60 + 0.461i)7-s + (−1.63 + 1.63i)8-s + (−2.80 − 1.06i)9-s + (−1.72 + 0.998i)10-s + (1.79 + 0.479i)11-s + (0.918 + 2.58i)12-s + (−2.76 − 2.31i)13-s + (−1.54 + 0.722i)14-s + (0.425 + 5.34i)15-s + (0.837 − 1.45i)16-s + (1.98 + 3.44i)17-s + ⋯
L(s)  = 1  + (0.440 − 0.118i)2-s + (0.181 − 0.983i)3-s + (−0.685 + 0.395i)4-s + (−1.33 + 0.358i)5-s + (−0.0362 − 0.454i)6-s + (−0.984 + 0.174i)7-s + (−0.578 + 0.578i)8-s + (−0.934 − 0.356i)9-s + (−0.546 + 0.315i)10-s + (0.540 + 0.144i)11-s + (0.265 + 0.746i)12-s + (−0.765 − 0.643i)13-s + (−0.413 + 0.193i)14-s + (0.109 + 1.37i)15-s + (0.209 − 0.362i)16-s + (0.482 + 0.835i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.939 - 0.341i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.939 - 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0144953 + 0.0823596i\)
\(L(\frac12)\) \(\approx\) \(0.0144953 + 0.0823596i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.313 + 1.70i)T \)
7 \( 1 + (2.60 - 0.461i)T \)
13 \( 1 + (2.76 + 2.31i)T \)
good2 \( 1 + (-0.623 + 0.167i)T + (1.73 - i)T^{2} \)
5 \( 1 + (2.98 - 0.801i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.79 - 0.479i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.98 - 3.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.93 + 7.22i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.29 + 2.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.80iT - 29T^{2} \)
31 \( 1 + (0.388 + 0.104i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (8.52 - 2.28i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (7.30 + 7.30i)T + 41iT^{2} \)
43 \( 1 - 2.15iT - 43T^{2} \)
47 \( 1 + (-0.515 - 1.92i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.46 - 0.845i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.21 + 1.93i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.418 + 0.724i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.34 + 0.895i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.94 - 1.94i)T + 71iT^{2} \)
73 \( 1 + (-0.241 + 0.902i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.833 - 1.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.2 - 10.2i)T + 83iT^{2} \)
89 \( 1 + (0.330 + 1.23i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.39 + 6.39i)T - 97iT^{2} \)
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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

−11.85748183115854570567673089546, −10.67442677934090143315361707564, −9.125378856780675649392186493919, −8.434559707196322175616634458920, −7.36817663645607361940317220316, −6.63334732655372839012247717260, −5.10623876988971742237944175956, −3.65167224608373793463274082767, −2.91908811614360694929520562652, −0.05356502502720641783485609085, 3.46269106994453378763284566063, 4.03908673143940599013980470196, 5.01627122029510493677682442696, 6.24737928419781451101375899448, 7.69443862730335279929920101401, 8.800662178100422283687023365500, 9.614037229262164003772970685025, 10.27993527260344930604733161355, 11.76962116405000768104230415768, 12.20700351960619383984480055607

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