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Properties

Label 2-273-91.89-c1-0-13
Degree $2$
Conductor $273$
Sign $0.888 - 0.457i$
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  + (1.12 + 1.12i)2-s + (0.866 + 0.5i)3-s + 0.509i·4-s + (−0.973 − 3.63i)5-s + (0.409 + 1.53i)6-s + (2.28 + 1.33i)7-s + (1.66 − 1.66i)8-s + (0.499 + 0.866i)9-s + (2.97 − 5.15i)10-s + (0.872 + 3.25i)11-s + (−0.254 + 0.440i)12-s + (−3.03 + 1.94i)13-s + (1.06 + 4.05i)14-s + (0.973 − 3.63i)15-s + 4.75·16-s − 3.33·17-s + ⋯
L(s)  = 1  + (0.791 + 0.791i)2-s + (0.499 + 0.288i)3-s + 0.254i·4-s + (−0.435 − 1.62i)5-s + (0.167 + 0.624i)6-s + (0.863 + 0.504i)7-s + (0.590 − 0.590i)8-s + (0.166 + 0.288i)9-s + (0.942 − 1.63i)10-s + (0.263 + 0.982i)11-s + (−0.0734 + 0.127i)12-s + (−0.842 + 0.538i)13-s + (0.284 + 1.08i)14-s + (0.251 − 0.938i)15-s + 1.18·16-s − 0.809·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.888 - 0.457i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.888 - 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11680 + 0.513167i\)
\(L(\frac12)\) \(\approx\) \(2.11680 + 0.513167i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 - 1.33i)T \)
13 \( 1 + (3.03 - 1.94i)T \)
good2 \( 1 + (-1.12 - 1.12i)T + 2iT^{2} \)
5 \( 1 + (0.973 + 3.63i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.872 - 3.25i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 3.33T + 17T^{2} \)
19 \( 1 + (-0.733 - 0.196i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 3.56iT - 23T^{2} \)
29 \( 1 + (1.42 + 2.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.90 + 2.38i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (8.01 - 8.01i)T - 37iT^{2} \)
41 \( 1 + (-6.84 - 1.83i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-10.9 - 6.31i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.32 + 0.356i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.59 + 6.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.23 + 2.23i)T + 59iT^{2} \)
61 \( 1 + (-0.902 + 0.521i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.85 + 0.765i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (7.00 - 1.87i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.559 + 2.08i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.54 - 9.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.51 - 1.51i)T - 83iT^{2} \)
89 \( 1 + (2.23 + 2.23i)T + 89iT^{2} \)
97 \( 1 + (3.23 + 12.0i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.41323763146052489199305131826, −11.26891951905095114470229897627, −9.728902834577808099609950148745, −9.001176708343027540479718219712, −8.019622621383167726787595064287, −7.12399803510518353135765806083, −5.57265012862513623076200842030, −4.58310325296008317472893513600, −4.35882243027021429091590363077, −1.82085414784283545296468566252, 2.17311345299113981405928074499, 3.25753202678447966573466552019, 4.02918781336817973062861246207, 5.58312811404146315314713784136, 7.29070450934222293795219989127, 7.54531634788305727051696124031, 8.927199283878914106882243223534, 10.71089493800684103387833686751, 10.82634933352467995572766651246, 11.74942029405809791911925487998

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