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Properties

Label 2-273-273.257-c1-0-15
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: $\chi_{273} (257, \cdot )$
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

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

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