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Properties

Label 2-273-91.73-c1-0-17
Degree $2$
Conductor $273$
Sign $-0.586 - 0.809i$
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  + (0.587 − 2.19i)2-s + (−0.866 − 0.5i)3-s + (−2.72 − 1.57i)4-s + (−1.78 − 0.477i)5-s + (−1.60 + 1.60i)6-s + (−2.58 − 0.565i)7-s + (−1.83 + 1.83i)8-s + (0.499 + 0.866i)9-s + (−2.09 + 3.62i)10-s + (0.733 + 2.73i)11-s + (1.57 + 2.72i)12-s + (1.91 − 3.05i)13-s + (−2.75 + 5.33i)14-s + (1.30 + 1.30i)15-s + (−0.201 − 0.348i)16-s + (−1.30 + 2.26i)17-s + ⋯
L(s)  = 1  + (0.415 − 1.54i)2-s + (−0.499 − 0.288i)3-s + (−1.36 − 0.786i)4-s + (−0.797 − 0.213i)5-s + (−0.654 + 0.654i)6-s + (−0.976 − 0.213i)7-s + (−0.648 + 0.648i)8-s + (0.166 + 0.288i)9-s + (−0.662 + 1.14i)10-s + (0.221 + 0.825i)11-s + (0.453 + 0.786i)12-s + (0.532 − 0.846i)13-s + (−0.736 + 1.42i)14-s + (0.337 + 0.337i)15-s + (−0.0503 − 0.0871i)16-s + (−0.316 + 0.548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.586 - 0.809i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.586 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306737 + 0.601108i\)
\(L(\frac12)\) \(\approx\) \(0.306737 + 0.601108i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 + 0.565i)T \)
13 \( 1 + (-1.91 + 3.05i)T \)
good2 \( 1 + (-0.587 + 2.19i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (1.78 + 0.477i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.733 - 2.73i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.30 - 2.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.68 + 2.05i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-6.09 + 3.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.00T + 29T^{2} \)
31 \( 1 + (2.15 + 8.03i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-5.55 - 1.48i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.97 + 4.97i)T - 41iT^{2} \)
43 \( 1 + 5.75iT - 43T^{2} \)
47 \( 1 + (2.23 - 8.35i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.51 + 7.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.16 - 0.311i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.80 + 1.04i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0372 + 0.00997i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.50 + 1.50i)T + 71iT^{2} \)
73 \( 1 + (2.83 - 0.758i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.66 - 2.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.45 - 5.45i)T - 83iT^{2} \)
89 \( 1 + (-0.808 + 3.01i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.47 + 7.47i)T - 97iT^{2} \)
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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

−11.24169397482070141259743829058, −10.73054387672003894258763803765, −9.798092957016438578730204733292, −8.674964565074828660470097038774, −7.28047580063364987237198947065, −6.11221118427390428787994064149, −4.53168566615961884954629682120, −3.81090256253922906446529779215, −2.36555021407490340010281181376, −0.47578232934533225899820413508, 3.52277516451840765626100720278, 4.46865502697714214100743461835, 5.79858476520067343366652810997, 6.51281140386286864074812089888, 7.27701667195432725603008453156, 8.558315289082109385416216606217, 9.241670321609810876956928873023, 10.80595013296908380517067700004, 11.55844452190636729920161144374, 12.80235963578097924191537835955

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