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Properties

Label 2-273-273.230-c1-0-3
Degree $2$
Conductor $273$
Sign $0.999 - 0.0436i$
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  + (−1.43 − 0.830i)2-s + (−1.23 − 1.21i)3-s + (0.378 + 0.655i)4-s − 1.88·5-s + (0.771 + 2.77i)6-s + (0.896 + 2.48i)7-s + 2.06i·8-s + (0.0593 + 2.99i)9-s + (2.71 + 1.56i)10-s + (−1.52 − 0.882i)11-s + (0.326 − 1.27i)12-s + (3.51 − 0.780i)13-s + (0.778 − 4.32i)14-s + (2.33 + 2.29i)15-s + (2.47 − 4.27i)16-s + (−1.02 − 1.78i)17-s + ⋯
L(s)  = 1  + (−1.01 − 0.587i)2-s + (−0.714 − 0.700i)3-s + (0.189 + 0.327i)4-s − 0.844·5-s + (0.315 + 1.13i)6-s + (0.338 + 0.940i)7-s + 0.729i·8-s + (0.0197 + 0.999i)9-s + (0.858 + 0.495i)10-s + (−0.460 − 0.266i)11-s + (0.0943 − 0.366i)12-s + (0.976 − 0.216i)13-s + (0.208 − 1.15i)14-s + (0.603 + 0.591i)15-s + (0.617 − 1.06i)16-s + (−0.249 − 0.432i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 - 0.0436i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.999 - 0.0436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410816 + 0.00897895i\)
\(L(\frac12)\) \(\approx\) \(0.410816 + 0.00897895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.23 + 1.21i)T \)
7 \( 1 + (-0.896 - 2.48i)T \)
13 \( 1 + (-3.51 + 0.780i)T \)
good2 \( 1 + (1.43 + 0.830i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 1.88T + 5T^{2} \)
11 \( 1 + (1.52 + 0.882i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.02 + 1.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.72 - 0.997i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.43 - 2.56i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.27 - 4.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.13iT - 31T^{2} \)
37 \( 1 + (-2.24 + 3.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.68 - 8.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.05 - 3.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.61T + 47T^{2} \)
53 \( 1 + 6.33iT - 53T^{2} \)
59 \( 1 + (-7.51 - 13.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.61 - 0.930i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.79 - 3.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.21 - 4.16i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.58iT - 73T^{2} \)
79 \( 1 + 3.51T + 79T^{2} \)
83 \( 1 + 5.69T + 83T^{2} \)
89 \( 1 + (-3.64 + 6.30i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.28 + 1.89i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.61956454481741389144315508289, −11.09848974988050273146043191271, −10.28300663001884325885113408582, −8.763219851862360492042886216926, −8.345108486661338186796622020109, −7.27785332072213668751856980717, −5.89038089054620529253254789939, −4.90721533689591290600873614705, −2.80391209535581598790261423229, −1.22767812453089315001440751982, 0.57589782506760920650438482002, 3.79848611040568970287086436020, 4.50330153832799593232265920127, 6.19157108809918922850093716560, 7.12470007240284675342695156389, 8.088109236508334299474669915265, 8.898087069481986263965735321564, 10.08558851137634284469481537355, 10.72461663531533109545080606705, 11.54318599694359651032340749239

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