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Properties

Label 2-273-273.272-c1-0-23
Degree $2$
Conductor $273$
Sign $0.895 - 0.444i$
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  + 2.35·2-s + (0.813 + 1.52i)3-s + 3.52·4-s − 0.514i·5-s + (1.91 + 3.59i)6-s + (−1.49 − 2.18i)7-s + 3.58·8-s + (−1.67 + 2.48i)9-s − 1.21i·10-s − 2.08·11-s + (2.86 + 5.39i)12-s + (−2.92 − 2.11i)13-s + (−3.50 − 5.13i)14-s + (0.787 − 0.418i)15-s + 1.38·16-s − 0.359·17-s + ⋯
L(s)  = 1  + 1.66·2-s + (0.469 + 0.882i)3-s + 1.76·4-s − 0.230i·5-s + (0.780 + 1.46i)6-s + (−0.563 − 0.826i)7-s + 1.26·8-s + (−0.558 + 0.829i)9-s − 0.382i·10-s − 0.629·11-s + (0.828 + 1.55i)12-s + (−0.809 − 0.586i)13-s + (−0.936 − 1.37i)14-s + (0.203 − 0.108i)15-s + 0.345·16-s − 0.0871·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.01080 + 0.705778i\)
\(L(\frac12)\) \(\approx\) \(3.01080 + 0.705778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.813 - 1.52i)T \)
7 \( 1 + (1.49 + 2.18i)T \)
13 \( 1 + (2.92 + 2.11i)T \)
good2 \( 1 - 2.35T + 2T^{2} \)
5 \( 1 + 0.514iT - 5T^{2} \)
11 \( 1 + 2.08T + 11T^{2} \)
17 \( 1 + 0.359T + 17T^{2} \)
19 \( 1 - 3.55T + 19T^{2} \)
23 \( 1 - 1.93iT - 23T^{2} \)
29 \( 1 - 8.77iT - 29T^{2} \)
31 \( 1 - 7.37T + 31T^{2} \)
37 \( 1 + 2.76iT - 37T^{2} \)
41 \( 1 + 5.37iT - 41T^{2} \)
43 \( 1 + 0.383T + 43T^{2} \)
47 \( 1 - 6.90iT - 47T^{2} \)
53 \( 1 + 9.21iT - 53T^{2} \)
59 \( 1 + 1.01iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + 3.71iT - 67T^{2} \)
71 \( 1 + 2.79T + 71T^{2} \)
73 \( 1 + 6.83T + 73T^{2} \)
79 \( 1 - 9.78T + 79T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 - 15.0iT - 89T^{2} \)
97 \( 1 + 8.34T + 97T^{2} \)
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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

−12.32883864862518022673429003278, −11.06288238050418648609820129947, −10.33493607888375243278640064348, −9.324985172045284594741773700326, −7.86322447595557768327187244886, −6.81353187348465641843350405164, −5.35859781832038845386216603160, −4.77894384651548322310758341569, −3.57485058589994722756476805405, −2.78942489401876401521255542613, 2.42331850791632652904629297311, 3.04080170915226791012617581791, 4.60880894774880695721972678762, 5.82689911017774797083793301731, 6.58845454467853081919147299679, 7.53477891040017855198731075613, 8.828446392752073391930203496980, 10.04396141511425319496242178242, 11.72018225207154316248235871646, 11.95190488848488099738678604999

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