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Properties

Label 2-273-91.41-c1-0-11
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: $\chi_{273} (223, \cdot )$
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

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

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