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Properties

Label 2-273-91.59-c1-0-6
Degree $2$
Conductor $273$
Sign $0.209 - 0.977i$
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  + (1.55 + 1.55i)2-s + (0.866 − 0.5i)3-s + 2.81i·4-s + (−0.926 − 0.248i)5-s + (2.12 + 0.568i)6-s + (0.619 + 2.57i)7-s + (−1.27 + 1.27i)8-s + (0.499 − 0.866i)9-s + (−1.05 − 1.82i)10-s + (2.73 + 0.732i)11-s + (1.40 + 2.44i)12-s + (−2.97 − 2.03i)13-s + (−3.03 + 4.95i)14-s + (−0.926 + 0.248i)15-s + 1.68·16-s − 5.07·17-s + ⋯
L(s)  = 1  + (1.09 + 1.09i)2-s + (0.499 − 0.288i)3-s + 1.40i·4-s + (−0.414 − 0.111i)5-s + (0.865 + 0.231i)6-s + (0.234 + 0.972i)7-s + (−0.449 + 0.449i)8-s + (0.166 − 0.288i)9-s + (−0.332 − 0.576i)10-s + (0.824 + 0.220i)11-s + (0.406 + 0.704i)12-s + (−0.825 − 0.564i)13-s + (−0.810 + 1.32i)14-s + (−0.239 + 0.0640i)15-s + 0.422·16-s − 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.209 - 0.977i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.209 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87208 + 1.51356i\)
\(L(\frac12)\) \(\approx\) \(1.87208 + 1.51356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.619 - 2.57i)T \)
13 \( 1 + (2.97 + 2.03i)T \)
good2 \( 1 + (-1.55 - 1.55i)T + 2iT^{2} \)
5 \( 1 + (0.926 + 0.248i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.73 - 0.732i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 5.07T + 17T^{2} \)
19 \( 1 + (0.114 + 0.429i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 3.37iT - 23T^{2} \)
29 \( 1 + (-4.91 + 8.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.861 - 3.21i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.72 - 3.72i)T - 37iT^{2} \)
41 \( 1 + (1.13 + 4.21i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.57 - 1.48i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.43 + 9.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.30 - 7.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.07 + 2.07i)T + 59iT^{2} \)
61 \( 1 + (-11.1 - 6.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.60 - 9.73i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.206 - 0.770i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.79 - 1.01i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.44 - 4.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.86 + 2.86i)T - 83iT^{2} \)
89 \( 1 + (-7.38 - 7.38i)T + 89iT^{2} \)
97 \( 1 + (4.64 + 1.24i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.24041994661238945267805465474, −11.80865597677722914569118156363, −10.08703052072307932029901541063, −8.773949742986085099706042000471, −8.073970963213751537772709589432, −6.98065652010645835996711898857, −6.18855245564962046653758782235, −4.95619802822815743282830293150, −4.05048612749179667300070954271, −2.51021515207014476167126155521, 1.79186812306608050984644025460, 3.31790691761404303615272505748, 4.15102992708667698550512705950, 4.93220598982988367398758043834, 6.66157870310951243799714462448, 7.76192998001449748360721366381, 9.114094484700046642458812098184, 10.11522226289016599507836950549, 11.07743652942579172549421035357, 11.60597576581889805346662280186

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