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Properties

Label 2-273-91.23-c1-0-17
Degree $2$
Conductor $273$
Sign $-0.691 - 0.722i$
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  − 2.22i·2-s + (−0.5 + 0.866i)3-s − 2.93·4-s + (−0.701 − 0.404i)5-s + (1.92 + 1.11i)6-s + (−2.44 − 1.00i)7-s + 2.08i·8-s + (−0.499 − 0.866i)9-s + (−0.900 + 1.55i)10-s + (−2.66 − 1.53i)11-s + (1.46 − 2.54i)12-s + (−3.01 + 1.97i)13-s + (−2.24 + 5.43i)14-s + (0.701 − 0.404i)15-s − 1.23·16-s + 2.79·17-s + ⋯
L(s)  = 1  − 1.57i·2-s + (−0.288 + 0.499i)3-s − 1.46·4-s + (−0.313 − 0.181i)5-s + (0.785 + 0.453i)6-s + (−0.924 − 0.381i)7-s + 0.738i·8-s + (−0.166 − 0.288i)9-s + (−0.284 + 0.492i)10-s + (−0.803 − 0.463i)11-s + (0.424 − 0.734i)12-s + (−0.836 + 0.548i)13-s + (−0.599 + 1.45i)14-s + (0.181 − 0.104i)15-s − 0.309·16-s + 0.678·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.691 - 0.722i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.164152 + 0.384145i\)
\(L(\frac12)\) \(\approx\) \(0.164152 + 0.384145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.44 + 1.00i)T \)
13 \( 1 + (3.01 - 1.97i)T \)
good2 \( 1 + 2.22iT - 2T^{2} \)
5 \( 1 + (0.701 + 0.404i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.66 + 1.53i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.79T + 17T^{2} \)
19 \( 1 + (3.73 - 2.15i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.95T + 23T^{2} \)
29 \( 1 + (2.84 + 4.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.93 + 1.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.72iT - 37T^{2} \)
41 \( 1 + (-8.48 + 4.90i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.85 - 4.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.31 + 3.06i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.83 - 4.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.93iT - 59T^{2} \)
61 \( 1 + (-1.51 - 2.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.59 + 4.96i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.84 + 4.52i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.75 - 1.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.13 + 1.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.52iT - 83T^{2} \)
89 \( 1 + 3.32iT - 89T^{2} \)
97 \( 1 + (-11.0 - 6.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.23232708728752843225502612398, −10.42677264597243330562153975387, −9.809133772450971815204581514037, −8.945054473530263246126704873385, −7.53054111884427595385243612701, −6.03511251457517246293242103651, −4.58578127109033662787115000520, −3.68481388264130666169483402917, −2.51903289299371847162521829748, −0.31384944671436732298957379366, 2.88112555195610081689085783149, 4.84716624217793236496917490082, 5.67784269137296858074833691298, 6.75806399028887709138605216374, 7.37081395186959741969794011537, 8.260956852201942051571741603516, 9.362734758071024209181718317719, 10.44058372617957934945704156176, 11.75363681369122281885376186365, 12.90181914760070035448989384837

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