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Properties

Label 2-273-91.45-c1-0-7
Degree $2$
Conductor $273$
Sign $0.421 - 0.906i$
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  + (−0.411 + 0.411i)2-s + (0.866 − 0.5i)3-s + 1.66i·4-s + (−0.180 + 0.672i)5-s + (−0.150 + 0.562i)6-s + (2.60 + 0.483i)7-s + (−1.50 − 1.50i)8-s + (0.499 − 0.866i)9-s + (−0.202 − 0.351i)10-s + (−0.230 + 0.859i)11-s + (0.830 + 1.43i)12-s + (0.659 + 3.54i)13-s + (−1.27 + 0.871i)14-s + (0.180 + 0.672i)15-s − 2.08·16-s − 0.0460·17-s + ⋯
L(s)  = 1  + (−0.291 + 0.291i)2-s + (0.499 − 0.288i)3-s + 0.830i·4-s + (−0.0806 + 0.300i)5-s + (−0.0615 + 0.229i)6-s + (0.983 + 0.182i)7-s + (−0.532 − 0.532i)8-s + (0.166 − 0.288i)9-s + (−0.0641 − 0.111i)10-s + (−0.0694 + 0.259i)11-s + (0.239 + 0.415i)12-s + (0.183 + 0.983i)13-s + (−0.339 + 0.233i)14-s + (0.0465 + 0.173i)15-s − 0.520·16-s − 0.0111·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.421 - 0.906i$
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: \((2,\ 273,\ (\ :1/2),\ 0.421 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12632 + 0.718337i\)
\(L(\frac12)\) \(\approx\) \(1.12632 + 0.718337i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.483i)T \)
13 \( 1 + (-0.659 - 3.54i)T \)
good2 \( 1 + (0.411 - 0.411i)T - 2iT^{2} \)
5 \( 1 + (0.180 - 0.672i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.230 - 0.859i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.0460T + 17T^{2} \)
19 \( 1 + (0.843 - 0.225i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 3.19iT - 23T^{2} \)
29 \( 1 + (-4.08 + 7.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.90 + 1.04i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.97 + 4.97i)T + 37iT^{2} \)
41 \( 1 + (8.56 - 2.29i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.29 + 0.746i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (12.1 + 3.24i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.89 + 8.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.25 + 6.25i)T - 59iT^{2} \)
61 \( 1 + (-0.877 - 0.506i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.42 - 0.648i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.798 + 0.213i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.09 + 15.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.73 + 8.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.50 + 3.50i)T + 83iT^{2} \)
89 \( 1 + (3.69 - 3.69i)T - 89iT^{2} \)
97 \( 1 + (0.288 - 1.07i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.89399235529175406371750740335, −11.45153442495584899286532265396, −9.996726978740414607989506833858, −8.866167464154108772962340984432, −8.232624411316549918578272117470, −7.33512351820544485338352122876, −6.49001106256758220975688106227, −4.74709539302512926693880079853, −3.52262490789708420022625297122, −2.07019222840527118883340870138, 1.25389518019138165525528248413, 2.86288906019461196015092642440, 4.60539090651213664845461585894, 5.39680971010398954239627056609, 6.82772907200620293678588765382, 8.431515211644939200544441050119, 8.583322235539919467621062349593, 10.09994887533310807539336383289, 10.55544643960715557164490029774, 11.48781999367559114147274512158

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