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Properties

Label 2-273-91.31-c1-0-0
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: $\chi_{273} (31, \cdot )$
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

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

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