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Properties

Label 2-273-21.20-c1-0-4
Degree $2$
Conductor $273$
Sign $-0.410 - 0.911i$
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  + 0.875i·2-s + (−1.71 − 0.210i)3-s + 1.23·4-s − 0.171·5-s + (0.184 − 1.50i)6-s + (−2.26 + 1.37i)7-s + 2.83i·8-s + (2.91 + 0.722i)9-s − 0.150i·10-s + 2.28i·11-s + (−2.12 − 0.259i)12-s + i·13-s + (−1.20 − 1.98i)14-s + (0.295 + 0.0360i)15-s − 0.0117·16-s + 1.19·17-s + ⋯
L(s)  = 1  + 0.619i·2-s + (−0.992 − 0.121i)3-s + 0.616·4-s − 0.0767·5-s + (0.0751 − 0.614i)6-s + (−0.855 + 0.518i)7-s + 1.00i·8-s + (0.970 + 0.240i)9-s − 0.0475i·10-s + 0.689i·11-s + (−0.612 − 0.0748i)12-s + 0.277i·13-s + (−0.320 − 0.529i)14-s + (0.0761 + 0.00931i)15-s − 0.00293·16-s + 0.290·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.410 - 0.911i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.410 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.501018 + 0.774953i\)
\(L(\frac12)\) \(\approx\) \(0.501018 + 0.774953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.210i)T \)
7 \( 1 + (2.26 - 1.37i)T \)
13 \( 1 - iT \)
good2 \( 1 - 0.875iT - 2T^{2} \)
5 \( 1 + 0.171T + 5T^{2} \)
11 \( 1 - 2.28iT - 11T^{2} \)
17 \( 1 - 1.19T + 17T^{2} \)
19 \( 1 - 7.90iT - 19T^{2} \)
23 \( 1 - 4.61iT - 23T^{2} \)
29 \( 1 + 8.57iT - 29T^{2} \)
31 \( 1 + 4.42iT - 31T^{2} \)
37 \( 1 + 3.27T + 37T^{2} \)
41 \( 1 - 8.03T + 41T^{2} \)
43 \( 1 - 7.41T + 43T^{2} \)
47 \( 1 - 5.85T + 47T^{2} \)
53 \( 1 + 4.72iT - 53T^{2} \)
59 \( 1 - 8.13T + 59T^{2} \)
61 \( 1 - 6.04iT - 61T^{2} \)
67 \( 1 - 2.30T + 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 4.51iT - 73T^{2} \)
79 \( 1 + 0.702T + 79T^{2} \)
83 \( 1 + 16.1T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 2.15iT - 97T^{2} \)
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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

−12.03402887290296964513021192593, −11.53643085125342814202624119886, −10.26682214040815380052284114124, −9.573346066886167440631111974097, −7.909154867846279732366048950351, −7.20715943673509684663637617276, −5.98032598777255381982090255040, −5.74006836657516509865709840505, −4.00425047011282243268301331567, −2.06527102295719129472589543654, 0.794282659688911662353314975245, 2.88679519206654390959579474444, 4.10979019853078245318178711665, 5.60742200664378765105054301978, 6.66363107812031064604727078648, 7.28530816784409751851164274465, 9.050538226451652766895532179418, 10.13193800074808534859523784126, 10.81008244995290727136427275290, 11.39065821332838912659774198397

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