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Properties

Label 2-273-91.73-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.522 - 0.852i$
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  + (−0.242 + 0.905i)2-s + (−0.866 − 0.5i)3-s + (0.970 + 0.560i)4-s + (−3.03 − 0.812i)5-s + (0.663 − 0.663i)6-s + (2.50 + 0.856i)7-s + (−2.06 + 2.06i)8-s + (0.499 + 0.866i)9-s + (1.47 − 2.55i)10-s + (1.00 + 3.76i)11-s + (−0.560 − 0.970i)12-s + (−1.80 + 3.12i)13-s + (−1.38 + 2.05i)14-s + (2.22 + 2.22i)15-s + (−0.251 − 0.435i)16-s + (−2.46 + 4.27i)17-s + ⋯
L(s)  = 1  + (−0.171 + 0.640i)2-s + (−0.499 − 0.288i)3-s + (0.485 + 0.280i)4-s + (−1.35 − 0.363i)5-s + (0.270 − 0.270i)6-s + (0.946 + 0.323i)7-s + (−0.731 + 0.731i)8-s + (0.166 + 0.288i)9-s + (0.465 − 0.806i)10-s + (0.304 + 1.13i)11-s + (−0.161 − 0.280i)12-s + (−0.500 + 0.865i)13-s + (−0.369 + 0.550i)14-s + (0.573 + 0.573i)15-s + (−0.0627 − 0.108i)16-s + (−0.598 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403995 + 0.721058i\)
\(L(\frac12)\) \(\approx\) \(0.403995 + 0.721058i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.50 - 0.856i)T \)
13 \( 1 + (1.80 - 3.12i)T \)
good2 \( 1 + (0.242 - 0.905i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (3.03 + 0.812i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.00 - 3.76i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.46 - 4.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.63 + 0.973i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-6.01 + 3.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.26T + 29T^{2} \)
31 \( 1 + (-1.74 - 6.50i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.34 + 1.69i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.12 + 4.12i)T - 41iT^{2} \)
43 \( 1 + 3.84iT - 43T^{2} \)
47 \( 1 + (0.516 - 1.92i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.345 + 0.598i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-13.2 + 3.53i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.50 + 4.91i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.16 - 2.18i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.76 + 1.76i)T + 71iT^{2} \)
73 \( 1 + (-14.4 + 3.85i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.69 + 4.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.16 - 6.16i)T - 83iT^{2} \)
89 \( 1 + (2.02 - 7.54i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.07 - 6.07i)T - 97iT^{2} \)
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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

−12.26724639753036636855171834855, −11.41758600558555810907039873209, −10.70955324465484599891550418131, −8.817802383717299372476266372058, −8.287800512366138152317924714086, −7.17638303147875762813965706959, −6.71121210736677489206994308644, −5.05440908657981840106353784355, −4.18499679821879901655204824520, −2.07564207739111426826909612856, 0.71277061292196961131939407370, 2.89206191476201179478139826446, 4.06086603161374166788977920604, 5.35491827901996193918561154700, 6.73339216225139640822886980915, 7.64612800979924990280411146585, 8.730013957665581863722127357698, 10.08937080556555372312503047642, 11.05465776383362360509328893070, 11.34146945309043795437947341749

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