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Properties

Label 2-273-7.2-c1-0-13
Degree $2$
Conductor $273$
Sign $-0.201 + 0.979i$
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  + (1.07 − 1.86i)2-s + (0.5 + 0.866i)3-s + (−1.31 − 2.26i)4-s + (1.86 − 3.23i)5-s + 2.14·6-s + (−1.88 + 1.86i)7-s − 1.33·8-s + (−0.499 + 0.866i)9-s + (−4.00 − 6.94i)10-s + (−0.264 − 0.458i)11-s + (1.31 − 2.26i)12-s − 13-s + (1.44 + 5.50i)14-s + 3.73·15-s + (1.18 − 2.05i)16-s + (1.90 + 3.29i)17-s + ⋯
L(s)  = 1  + (0.759 − 1.31i)2-s + (0.288 + 0.499i)3-s + (−0.655 − 1.13i)4-s + (0.834 − 1.44i)5-s + 0.877·6-s + (−0.710 + 0.703i)7-s − 0.471·8-s + (−0.166 + 0.288i)9-s + (−1.26 − 2.19i)10-s + (−0.0797 − 0.138i)11-s + (0.378 − 0.655i)12-s − 0.277·13-s + (0.386 + 1.47i)14-s + 0.963·15-s + (0.296 − 0.513i)16-s + (0.461 + 0.799i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.201 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32256 - 1.62310i\)
\(L(\frac12)\) \(\approx\) \(1.32256 - 1.62310i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.88 - 1.86i)T \)
13 \( 1 + T \)
good2 \( 1 + (-1.07 + 1.86i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.86 + 3.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.264 + 0.458i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.90 - 3.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.94 - 3.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.21 + 2.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.56T + 29T^{2} \)
31 \( 1 + (-2.49 - 4.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.72 - 8.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.21T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + (5.77 - 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.81 + 6.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.73 - 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.90 + 5.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.07 + 5.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.88T + 71T^{2} \)
73 \( 1 + (3.22 + 5.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.85 + 3.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.16T + 83T^{2} \)
89 \( 1 + (-0.731 + 1.26i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−11.98161319242175247798345875313, −10.62936900994714144594424036082, −9.861678022291079521633576305921, −9.168769051490455964207166971207, −8.219388847137253557034357331965, −6.01798273864407470356222802809, −5.18784593616819562493441451580, −4.20509359181360605560377205314, −2.91510953548382236605816799337, −1.61984377270355520414216959297, 2.58382220764999486622474315019, 3.84085839662125509015534226972, 5.47881276287979001924190014175, 6.39637946727176890689279349036, 7.12138882484550870130280627010, 7.55906334501153790867236156358, 9.247195808145492291195680989615, 10.21283684855804112925432486475, 11.18993001354791495987815465735, 12.74578167049017096261835288186

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