Properties

Label 2-273-273.17-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.555 - 0.831i$
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  − 1.52·2-s + (−1.64 + 0.547i)3-s + 0.314·4-s + (1.84 − 1.06i)5-s + (2.49 − 0.832i)6-s + (−2.46 + 0.954i)7-s + 2.56·8-s + (2.40 − 1.79i)9-s + (−2.81 + 1.62i)10-s + (−0.983 − 1.70i)11-s + (−0.516 + 0.172i)12-s + (1.00 + 3.46i)13-s + (3.75 − 1.45i)14-s + (−2.45 + 2.76i)15-s − 4.53·16-s − 2.98·17-s + ⋯
L(s)  = 1  − 1.07·2-s + (−0.948 + 0.316i)3-s + 0.157·4-s + (0.826 − 0.477i)5-s + (1.02 − 0.340i)6-s + (−0.932 + 0.360i)7-s + 0.906·8-s + (0.800 − 0.599i)9-s + (−0.888 + 0.513i)10-s + (−0.296 − 0.513i)11-s + (−0.149 + 0.0497i)12-s + (0.278 + 0.960i)13-s + (1.00 − 0.388i)14-s + (−0.633 + 0.713i)15-s − 1.13·16-s − 0.722·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.131179 + 0.245341i\)
\(L(\frac12)\) \(\approx\) \(0.131179 + 0.245341i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.64 - 0.547i)T \)
7 \( 1 + (2.46 - 0.954i)T \)
13 \( 1 + (-1.00 - 3.46i)T \)
good2 \( 1 + 1.52T + 2T^{2} \)
5 \( 1 + (-1.84 + 1.06i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.983 + 1.70i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 2.98T + 17T^{2} \)
19 \( 1 + (1.80 - 3.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.06iT - 23T^{2} \)
29 \( 1 + (5.67 + 3.27i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.68 - 4.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.66iT - 37T^{2} \)
41 \( 1 + (-1.26 - 0.727i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.823 - 1.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.14 + 4.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.312 + 0.180i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 + (1.10 + 0.640i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.03 - 2.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.36 + 11.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.00 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.35 + 5.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.817iT - 83T^{2} \)
89 \( 1 - 15.4iT - 89T^{2} \)
97 \( 1 + (-9.24 - 16.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.00211022228565864514893298590, −11.01778061793488930531382773169, −10.13023716641089720330700229168, −9.342067735298909071601900911022, −8.916793542915183940525869140834, −7.35857066319022352667201585709, −6.20216628226476882481247876243, −5.38450865509861416189682026671, −3.96793859371719836924070092426, −1.61697321310669673362900207681, 0.34272990856459926108795275743, 2.26565220026399144790038707475, 4.41491371382666191960777136379, 5.81014555732781398972346295963, 6.75276459013622081741102537907, 7.53862179182709950186740834702, 8.897210700046326244631058138383, 9.884248505935597796438459840188, 10.51713073796150145251263830057, 11.03611727485657512938895591668

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