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Properties

Label 2-273-273.200-c1-0-22
Degree $2$
Conductor $273$
Sign $0.748 + 0.663i$
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  + (0.866 − 1.5i)3-s + (1.73 + i)4-s + (0.5 − 2.59i)7-s + (−1.5 − 2.59i)9-s + (3 − 1.73i)12-s + (−0.866 + 3.5i)13-s + (1.99 + 3.46i)16-s + (−1.23 − 0.330i)19-s + (−3.46 − 3i)21-s + (4.33 + 2.5i)25-s − 5.19·27-s + (3.46 − 4i)28-s + (−2.86 − 10.6i)31-s − 6i·36-s + (−2.40 + 8.96i)37-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (0.866 + 0.5i)4-s + (0.188 − 0.981i)7-s + (−0.5 − 0.866i)9-s + (0.866 − 0.499i)12-s + (−0.240 + 0.970i)13-s + (0.499 + 0.866i)16-s + (−0.282 − 0.0757i)19-s + (−0.755 − 0.654i)21-s + (0.866 + 0.5i)25-s − 1.00·27-s + (0.654 − 0.755i)28-s + (−0.514 − 1.92i)31-s − i·36-s + (−0.394 + 1.47i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.748 + 0.663i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.748 + 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61766 - 0.614127i\)
\(L(\frac12)\) \(\approx\) \(1.61766 - 0.614127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 1.5i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (0.866 - 3.5i)T \)
good2 \( 1 + (-1.73 - i)T^{2} \)
5 \( 1 + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.23 + 0.330i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (2.86 + 10.6i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.40 - 8.96i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 41iT^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.46 - 6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.205 - 0.767i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 71iT^{2} \)
73 \( 1 + (15.4 - 4.13i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.06 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.0717 + 0.0717i)T - 97iT^{2} \)
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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

−11.70001205060572290706203760056, −11.19903684803345788404414984717, −9.894906298134057226113959307038, −8.666686229138527669370415668387, −7.67632174342558231133387800131, −7.04849407572659673790749751181, −6.20667821347275736761351383920, −4.25880083933450224478574868539, −2.96980371316808273796371616269, −1.61023001958516016286348661855, 2.20513457695094955499927868300, 3.27393135133182694244412332112, 5.03176937330993522126408462253, 5.73926421859207841314585106419, 7.13188597114265530636648900114, 8.353078646265069113477089438741, 9.147179441100633216981922985294, 10.34917130005651616853333308230, 10.76271192307170268882449048859, 11.92232093257829574765379571428

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