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Properties

Label 2-273-13.10-c1-0-3
Degree $2$
Conductor $273$
Sign $0.242 - 0.970i$
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  + (−1.54 + 0.889i)2-s + (−0.5 − 0.866i)3-s + (0.582 − 1.00i)4-s + 0.681i·5-s + (1.54 + 0.889i)6-s + (0.866 + 0.5i)7-s − 1.48i·8-s + (−0.499 + 0.866i)9-s + (−0.606 − 1.05i)10-s + (−0.511 + 0.295i)11-s − 1.16·12-s + (3.45 − 1.03i)13-s − 1.77·14-s + (0.590 − 0.340i)15-s + (2.48 + 4.30i)16-s + (−3.14 + 5.44i)17-s + ⋯
L(s)  = 1  + (−1.08 + 0.629i)2-s + (−0.288 − 0.499i)3-s + (0.291 − 0.504i)4-s + 0.304i·5-s + (0.629 + 0.363i)6-s + (0.327 + 0.188i)7-s − 0.524i·8-s + (−0.166 + 0.288i)9-s + (−0.191 − 0.332i)10-s + (−0.154 + 0.0889i)11-s − 0.336·12-s + (0.957 − 0.287i)13-s − 0.475·14-s + (0.152 − 0.0880i)15-s + (0.621 + 1.07i)16-s + (−0.762 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.242 - 0.970i$
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: \((2,\ 273,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.505636 + 0.394869i\)
\(L(\frac12)\) \(\approx\) \(0.505636 + 0.394869i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.45 + 1.03i)T \)
good2 \( 1 + (1.54 - 0.889i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 0.681iT - 5T^{2} \)
11 \( 1 + (0.511 - 0.295i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.14 - 5.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.10 - 1.79i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.84 - 3.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.22 - 5.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.30iT - 31T^{2} \)
37 \( 1 + (9.12 - 5.26i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.136 + 0.0787i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.36 + 4.09i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.2iT - 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + (2.44 + 1.41i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.45 - 2.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.48 + 3.74i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.05 - 3.49i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.30iT - 73T^{2} \)
79 \( 1 + 3.33T + 79T^{2} \)
83 \( 1 + 13.6iT - 83T^{2} \)
89 \( 1 + (10.6 - 6.12i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.18 + 5.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.06955816212330573628240899644, −10.81512218413843955499140699461, −10.28629101467921073006856250813, −8.771560322585865834559037337562, −8.418398058126957856013035298700, −7.21789236165751446897095373495, −6.53945674636936381621570568418, −5.36292458814268005653358502364, −3.53644396979686716290642718205, −1.42803683863734388698153027173, 0.841896706745642262145486861644, 2.67010648588185920453983947084, 4.42215056466928688876868233540, 5.46936274722017873550196065236, 6.96705573764341064539330542499, 8.268221994620406261328772580716, 9.046107332586196802158201865010, 9.730667692525087037931233761302, 10.89863717589719635968057935676, 11.21573148249267888158598874876

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