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Properties

Label 2-273-13.10-c1-0-6
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: $\chi_{273} (127, \cdot )$
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

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

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