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Properties

Label 2-273-13.9-c1-0-3
Degree $2$
Conductor $273$
Sign $0.329 - 0.944i$
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 + 3.53·5-s + (−0.173 − 0.300i)6-s + (0.5 + 0.866i)7-s − 1.34·8-s + (−0.499 − 0.866i)9-s + (−0.613 + 1.06i)10-s + (1.64 − 2.84i)11-s − 1.87·12-s + (−2.99 − 2.01i)13-s − 0.347·14-s + (−1.76 + 3.05i)15-s + (−1.64 + 2.84i)16-s + (2.58 + 4.47i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.212i)2-s + (−0.288 + 0.499i)3-s + (0.469 + 0.813i)4-s + 1.57·5-s + (−0.0708 − 0.122i)6-s + (0.188 + 0.327i)7-s − 0.476·8-s + (−0.166 − 0.288i)9-s + (−0.193 + 0.335i)10-s + (0.496 − 0.859i)11-s − 0.542·12-s + (−0.830 − 0.557i)13-s − 0.0928·14-s + (−0.455 + 0.789i)15-s + (−0.411 + 0.712i)16-s + (0.626 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.329 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21535 + 0.862664i\)
\(L(\frac12)\) \(\approx\) \(1.21535 + 0.862664i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (2.99 + 2.01i)T \)
good2 \( 1 + (0.173 - 0.300i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.53T + 5T^{2} \)
11 \( 1 + (-1.64 + 2.84i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.58 - 4.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.03 + 5.25i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.745 - 1.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.36 - 4.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.59T + 31T^{2} \)
37 \( 1 + (3.95 - 6.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.82 + 10.0i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.19 + 10.7i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.162T + 47T^{2} \)
53 \( 1 - 6.84T + 53T^{2} \)
59 \( 1 + (5.80 + 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.911 + 1.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.59 - 7.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.24 - 3.88i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 + 9.22T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (0.0320 - 0.0555i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.224 + 0.388i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.14166433951647577858328755165, −11.00385566063454428903242739943, −10.24798229589735386069322287018, −9.137561503764069528683912748392, −8.476411710109050889562439114196, −6.99810289667231070750785367619, −6.03775211735957988400157913614, −5.23478329020457030749539254421, −3.48515001727778888681026929082, −2.16695683600031131614716884452, 1.51736990681486145729814613611, 2.36773519230931644309721154281, 4.77985039221304692796907112016, 5.84045611722474095731263250603, 6.56959956995091288060156776631, 7.55399978997009230190858076081, 9.377699108318907905347426081758, 9.788887718650282428611055104597, 10.62352475687715108874170085952, 11.74233085627031285800646406202

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