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Properties

Label 2-273-91.20-c1-0-5
Degree $2$
Conductor $273$
Sign $0.806 - 0.591i$
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 + (−1.54 − 2.15i)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 + (−0.582 − 0.812i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.806 - 0.591i$
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: \((2,\ 273,\ (\ :1/2),\ 0.806 - 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525109 + 0.171849i\)
\(L(\frac12)\) \(\approx\) \(0.525109 + 0.171849i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.54 + 2.15i)T \)
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

−11.96398271983442222095366655251, −10.76192474291673117163269198261, −9.937578480372922656102066152150, −9.387606802484963865828630871276, −7.70244003288681732921544343317, −7.48677318521876528164903442069, −6.40065621502931657574258390425, −4.82862434598174380396714263476, −3.52736377866913461215447851604, −1.12360393644185662841011148354, 0.837004674749912805691950285264, 2.99794300214478659227977743313, 4.67933040549707851465461064112, 5.75620667181200931792111973798, 6.91001749053784806160536777615, 8.422914887583438312347700207174, 9.027252676330472915147621346506, 9.722653516789797427583523622498, 10.87570843105895726530618541143, 11.55958716875846399068114030048

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