Properties

Label 2-273-13.4-c1-0-1
Degree $2$
Conductor $273$
Sign $0.986 - 0.165i$
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  + (−2.25 − 1.30i)2-s + (−0.5 + 0.866i)3-s + (2.40 + 4.15i)4-s − 1.50i·5-s + (2.25 − 1.30i)6-s + (−0.866 + 0.5i)7-s − 7.30i·8-s + (−0.499 − 0.866i)9-s + (−1.96 + 3.39i)10-s + (0.0753 + 0.0434i)11-s − 4.80·12-s + (−1.30 + 3.36i)13-s + 2.60·14-s + (1.30 + 0.752i)15-s + (−4.72 + 8.18i)16-s + (3.24 + 5.62i)17-s + ⋯
L(s)  = 1  + (−1.59 − 0.922i)2-s + (−0.288 + 0.499i)3-s + (1.20 + 2.07i)4-s − 0.673i·5-s + (0.922 − 0.532i)6-s + (−0.327 + 0.188i)7-s − 2.58i·8-s + (−0.166 − 0.288i)9-s + (−0.620 + 1.07i)10-s + (0.0227 + 0.0131i)11-s − 1.38·12-s + (−0.361 + 0.932i)13-s + 0.697·14-s + (0.336 + 0.194i)15-s + (−1.18 + 2.04i)16-s + (0.787 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.986 - 0.165i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.986 - 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.490151 + 0.0409177i\)
\(L(\frac12)\) \(\approx\) \(0.490151 + 0.0409177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (1.30 - 3.36i)T \)
good2 \( 1 + (2.25 + 1.30i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 1.50iT - 5T^{2} \)
11 \( 1 + (-0.0753 - 0.0434i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.24 - 5.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.58 + 2.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.60 - 4.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.98 + 5.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.00iT - 31T^{2} \)
37 \( 1 + (-8.24 - 4.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.57 - 3.21i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.40 - 5.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.83iT - 47T^{2} \)
53 \( 1 + 3.28T + 53T^{2} \)
59 \( 1 + (2.31 - 1.33i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.27 - 5.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.18 + 4.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.1 - 6.45i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.5iT - 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 - 1.29iT - 83T^{2} \)
89 \( 1 + (-13.7 - 7.92i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.08 - 2.93i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.74967548827776552999295846883, −10.84357067047201607727327422666, −9.753931775434619153373476844399, −9.439966759837404089321934197267, −8.422005754308995730410405867661, −7.50333530257820667217605992517, −6.08043244021769634446644597175, −4.37738897376545960411673567704, −2.97811001665262521775832877720, −1.28555986199851768307966132343, 0.76256831023569140720676677006, 2.75750080339041911404455371521, 5.36751854056831681813348179284, 6.27531233003720376955749483959, 7.34980120897225293911011025538, 7.64674166030520453263691651434, 8.929761086264582875496982915115, 9.965444994188608302004283346538, 10.50953931997516293790961612109, 11.54124865082904266836305440488

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