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Properties

Label 2-273-91.47-c1-0-9
Degree $2$
Conductor $273$
Sign $-0.551 + 0.834i$
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  + (−0.812 + 0.217i)2-s + (−0.866 + 0.5i)3-s + (−1.11 + 0.646i)4-s + (0.429 + 1.60i)5-s + (0.594 − 0.594i)6-s + (−2.64 − 0.0720i)7-s + (1.95 − 1.95i)8-s + (0.499 − 0.866i)9-s + (−0.697 − 1.20i)10-s + (−3.88 − 1.04i)11-s + (0.646 − 1.11i)12-s + (3.57 + 0.487i)13-s + (2.16 − 0.517i)14-s + (−1.17 − 1.17i)15-s + (0.128 − 0.222i)16-s + (−3.34 − 5.78i)17-s + ⋯
L(s)  = 1  + (−0.574 + 0.153i)2-s + (−0.499 + 0.288i)3-s + (−0.559 + 0.323i)4-s + (0.191 + 0.716i)5-s + (0.242 − 0.242i)6-s + (−0.999 − 0.0272i)7-s + (0.692 − 0.692i)8-s + (0.166 − 0.288i)9-s + (−0.220 − 0.381i)10-s + (−1.17 − 0.313i)11-s + (0.186 − 0.323i)12-s + (0.990 + 0.135i)13-s + (0.578 − 0.138i)14-s + (−0.302 − 0.302i)15-s + (0.0320 − 0.0555i)16-s + (−0.810 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.551 + 0.834i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.551 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0254532 - 0.0473138i\)
\(L(\frac12)\) \(\approx\) \(0.0254532 - 0.0473138i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.0720i)T \)
13 \( 1 + (-3.57 - 0.487i)T \)
good2 \( 1 + (0.812 - 0.217i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.429 - 1.60i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.88 + 1.04i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.34 + 5.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.110 - 0.410i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.27 + 2.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.18T + 29T^{2} \)
31 \( 1 + (6.32 + 1.69i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.220 - 0.821i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.80 + 1.80i)T - 41iT^{2} \)
43 \( 1 - 9.35iT - 43T^{2} \)
47 \( 1 + (-1.41 + 0.378i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.75 - 3.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0628 - 0.234i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (11.0 + 6.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.20 - 8.23i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.91 + 1.91i)T + 71iT^{2} \)
73 \( 1 + (1.91 - 7.13i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.0938 - 0.162i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.12 - 4.12i)T - 83iT^{2} \)
89 \( 1 + (7.56 - 2.02i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.202 - 0.202i)T - 97iT^{2} \)
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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

−11.25418313464484568335598997243, −10.53360479708463692124759831928, −9.667316204417275008421132492507, −8.899677087312317672363252267939, −7.64536584995009983682440494843, −6.70256471234975390747560877083, −5.62702128550461720043955946031, −4.18390853625984500586091410374, −2.94083642296905532344213538055, −0.05215878454330270005935296323, 1.76220730853862267666122633077, 3.94436772412648813236281106326, 5.33360684531598694465752746847, 6.04332561512102912286816189846, 7.52483123867999309821029853585, 8.609154723432049297347904901937, 9.321415173088547365085689290562, 10.41741355418469761458049806482, 10.90555489559018624791247031172, 12.42774167294168248379609360134

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