Properties

Label 2-273-91.20-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.422 - 0.906i$
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  + (−1.38 + 0.369i)2-s + (0.866 + 0.5i)3-s + (0.0377 − 0.0217i)4-s + (0.512 − 0.512i)5-s + (−1.38 − 0.369i)6-s + (−2.41 + 1.09i)7-s + (1.97 − 1.97i)8-s + (0.499 + 0.866i)9-s + (−0.517 + 0.896i)10-s + (1.38 + 5.18i)11-s + 0.0435·12-s + (−0.0545 + 3.60i)13-s + (2.92 − 2.39i)14-s + (0.699 − 0.187i)15-s + (−2.04 + 3.53i)16-s + (−1.31 − 2.28i)17-s + ⋯
L(s)  = 1  + (−0.976 + 0.261i)2-s + (0.499 + 0.288i)3-s + (0.0188 − 0.0108i)4-s + (0.229 − 0.229i)5-s + (−0.563 − 0.151i)6-s + (−0.910 + 0.412i)7-s + (0.699 − 0.699i)8-s + (0.166 + 0.288i)9-s + (−0.163 + 0.283i)10-s + (0.419 + 1.56i)11-s + 0.0125·12-s + (−0.0151 + 0.999i)13-s + (0.781 − 0.641i)14-s + (0.180 − 0.0484i)15-s + (−0.510 + 0.884i)16-s + (−0.319 − 0.553i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.422 - 0.906i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.422 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370450 + 0.581253i\)
\(L(\frac12)\) \(\approx\) \(0.370450 + 0.581253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 - 1.09i)T \)
13 \( 1 + (0.0545 - 3.60i)T \)
good2 \( 1 + (1.38 - 0.369i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.512 + 0.512i)T - 5iT^{2} \)
11 \( 1 + (-1.38 - 5.18i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.31 + 2.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.26 + 1.41i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.51 - 3.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.300 + 0.520i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.22 - 6.22i)T - 31iT^{2} \)
37 \( 1 + (-0.172 - 0.644i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.11 + 7.88i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-4.10 + 2.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.25 - 4.25i)T + 47iT^{2} \)
53 \( 1 - 0.282T + 53T^{2} \)
59 \( 1 + (1.21 - 4.54i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-13.0 + 7.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.48 + 1.20i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.11 + 15.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.04 - 3.04i)T + 73iT^{2} \)
79 \( 1 - 4.77T + 79T^{2} \)
83 \( 1 + (-2.42 + 2.42i)T - 83iT^{2} \)
89 \( 1 + (5.75 - 1.54i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (15.7 + 4.21i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.40624590500927649063623455445, −10.92753071423266518824179165956, −9.825989030255255062127044628608, −9.155671869169116527578807896000, −8.890825677778257407968424996205, −7.26515494805789726786515843676, −6.80111512815170285031012635680, −4.93960984078401077555748948761, −3.80902898031688993387746657762, −1.98966959376635457450061253763, 0.69924064443477109624551515402, 2.60069351030331694382364629880, 3.92964565829740952421758317514, 5.79941903465750468501223761369, 6.78256745833195086964460140808, 8.148783776387827258801987917782, 8.675999876484197709813828858365, 9.659705154949201108774680191952, 10.55204338787701425323701660291, 11.12770812108983935951527330562

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