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Properties

Label 2-273-273.41-c2-0-30
Degree $2$
Conductor $273$
Sign $0.777 - 0.628i$
Analytic cond. $7.43871$
Root an. cond. $2.72740$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.59 + 1.5i)3-s + (3.46 + 2i)4-s + (5.5 − 4.33i)7-s + (4.5 − 7.79i)9-s − 12·12-s + (12.9 − 0.5i)13-s + (7.99 + 13.8i)16-s + (−1.16 + 0.313i)19-s + (−7.79 + 19.5i)21-s + 25i·25-s + 27i·27-s + (27.7 − 4i)28-s + (36.8 + 36.8i)31-s + (31.1 − 18i)36-s + (7.92 − 29.5i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)4-s + (0.785 − 0.618i)7-s + (0.5 − 0.866i)9-s − 12-s + (0.999 − 0.0384i)13-s + (0.499 + 0.866i)16-s + (−0.0615 + 0.0164i)19-s + (−0.371 + 0.928i)21-s + i·25-s + i·27-s + (0.989 − 0.142i)28-s + (1.18 + 1.18i)31-s + (0.866 − 0.5i)36-s + (0.214 − 0.798i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.777 - 0.628i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ 0.777 - 0.628i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.58246 + 0.559443i\)
\(L(\frac12)\) \(\approx\) \(1.58246 + 0.559443i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.59 - 1.5i)T \)
7 \( 1 + (-5.5 + 4.33i)T \)
13 \( 1 + (-12.9 + 0.5i)T \)
good2 \( 1 + (-3.46 - 2i)T^{2} \)
5 \( 1 - 25iT^{2} \)
11 \( 1 + (104. + 60.5i)T^{2} \)
17 \( 1 + (144.5 + 250. i)T^{2} \)
19 \( 1 + (1.16 - 0.313i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-36.8 - 36.8i)T + 961iT^{2} \)
37 \( 1 + (-7.92 + 29.5i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-19.5 - 11.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 2.20e3iT^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (40.7 + 23.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (99.9 + 26.7i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-56.7 + 56.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 157.T + 6.24e3T^{2} \)
83 \( 1 - 6.88e3iT^{2} \)
89 \( 1 + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-180. + 48.3i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.55557743666062237999890683128, −10.94884351384107005414694340514, −10.31504176360114798993572390039, −8.881870159051292520493077224215, −7.73913310295526890346527412109, −6.78666694729203957919921063555, −5.80158831391029396399682555142, −4.51162442514737500719795672947, −3.39314380003941413088091634244, −1.37351331030314226692353512076, 1.17913949813971646103225868974, 2.42918784787568936222757426153, 4.58865866039001420005305636183, 5.80274204105946747741395112980, 6.34170575358964858112983512316, 7.54442273238665328706265655025, 8.487406478409136833640609147570, 10.01422738305869900304978565958, 10.88328294872887173252595998588, 11.58995156078726774174962042580

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