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Properties

Label 2-273-7.2-c1-0-5
Degree $2$
Conductor $273$
Sign $0.296 - 0.954i$
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  + (0.173 − 0.300i)2-s + (0.5 + 0.866i)3-s + (0.939 + 1.62i)4-s + (−0.326 + 0.565i)5-s + 0.347·6-s + (−2.05 + 1.66i)7-s + 1.34·8-s + (−0.499 + 0.866i)9-s + (0.113 + 0.196i)10-s + (−0.266 − 0.460i)11-s + (−0.939 + 1.62i)12-s + 13-s + (0.145 + 0.907i)14-s − 0.652·15-s + (−1.64 + 2.84i)16-s + (−0.560 − 0.970i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.212i)2-s + (0.288 + 0.499i)3-s + (0.469 + 0.813i)4-s + (−0.145 + 0.252i)5-s + 0.141·6-s + (−0.775 + 0.630i)7-s + 0.476·8-s + (−0.166 + 0.288i)9-s + (0.0358 + 0.0620i)10-s + (−0.0802 − 0.138i)11-s + (−0.271 + 0.469i)12-s + 0.277·13-s + (0.0388 + 0.242i)14-s − 0.168·15-s + (−0.411 + 0.712i)16-s + (−0.135 − 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.296 - 0.954i$
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: \((2,\ 273,\ (\ :1/2),\ 0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19292 + 0.878500i\)
\(L(\frac12)\) \(\approx\) \(1.19292 + 0.878500i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 - 1.66i)T \)
13 \( 1 - T \)
good2 \( 1 + (-0.173 + 0.300i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.326 - 0.565i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.266 + 0.460i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.560 + 0.970i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.152 - 0.264i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.00 + 6.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.78T + 29T^{2} \)
31 \( 1 + (0.294 + 0.509i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.43 + 2.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.04T + 41T^{2} \)
43 \( 1 - 2.94T + 43T^{2} \)
47 \( 1 + (-2.06 + 3.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.26 - 2.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.19 + 12.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.92 - 6.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.33 + 4.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.92T + 71T^{2} \)
73 \( 1 + (5.81 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.46 - 2.53i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.46T + 83T^{2} \)
89 \( 1 + (5.49 - 9.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−12.15920154037025580940819901160, −11.12039925095858321085011637829, −10.38009137394890463494845047214, −9.110376380575321231881207774483, −8.410155859078793777483769898026, −7.17810059438736441355742401884, −6.22492053912946405691276892648, −4.66976418244345027695388234427, −3.36326774071587102728682926231, −2.59059504175149591801815944807, 1.17507455901818428429550140513, 2.93351789920650251845436177056, 4.50563588954114942163101744582, 5.88513707041164825195723053751, 6.76545263227777183016467756879, 7.55632996601080875364379862019, 8.845398415030480255114966921359, 9.894374238523594599880735385857, 10.69516423836760826712654364642, 11.73670511068035167172931214343

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