Properties

Label 2-273-91.80-c1-0-12
Degree $2$
Conductor $273$
Sign $0.816 - 0.576i$
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  + (0.569 + 2.12i)2-s i·3-s + (−2.46 + 1.42i)4-s + (0.837 − 3.12i)5-s + (2.12 − 0.569i)6-s + (0.780 − 2.52i)7-s + (−1.31 − 1.31i)8-s − 9-s + 7.12·10-s + (2.85 + 2.85i)11-s + (1.42 + 2.46i)12-s + (2.61 + 2.47i)13-s + (5.81 + 0.218i)14-s + (−3.12 − 0.837i)15-s + (−0.796 + 1.38i)16-s + (−3.80 − 6.59i)17-s + ⋯
L(s)  = 1  + (0.402 + 1.50i)2-s − 0.577i·3-s + (−1.23 + 0.711i)4-s + (0.374 − 1.39i)5-s + (0.868 − 0.232i)6-s + (0.294 − 0.955i)7-s + (−0.465 − 0.465i)8-s − 0.333·9-s + 2.25·10-s + (0.859 + 0.859i)11-s + (0.410 + 0.711i)12-s + (0.726 + 0.687i)13-s + (1.55 + 0.0584i)14-s + (−0.806 − 0.216i)15-s + (−0.199 + 0.345i)16-s + (−0.922 − 1.59i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.816 - 0.576i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.816 - 0.576i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58206 + 0.502170i\)
\(L(\frac12)\) \(\approx\) \(1.58206 + 0.502170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (-0.780 + 2.52i)T \)
13 \( 1 + (-2.61 - 2.47i)T \)
good2 \( 1 + (-0.569 - 2.12i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.837 + 3.12i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.85 - 2.85i)T + 11iT^{2} \)
17 \( 1 + (3.80 + 6.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.95 - 1.95i)T + 19iT^{2} \)
23 \( 1 + (3.89 + 2.24i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.49 - 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.58 + 0.691i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.64 - 0.439i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.95 - 11.0i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.30 - 3.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.48 + 0.666i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.91 - 6.77i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.90 + 2.11i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 9.20iT - 61T^{2} \)
67 \( 1 + (-5.52 + 5.52i)T - 67iT^{2} \)
71 \( 1 + (0.721 + 2.69i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.453 - 1.69i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.31 - 7.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.63 + 1.63i)T + 83iT^{2} \)
89 \( 1 + (4.66 + 17.4i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.7 - 3.41i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.32312241412088168145114548915, −11.33405971600472215527899348126, −9.614231828983135025037697354170, −8.815410199084299035111539757423, −7.88504893936149912300852350155, −6.96056129598381914714124891252, −6.20336981184030573727950018179, −4.83647669560884578614793870815, −4.33722005252205877186705653922, −1.43513930230917192781230554531, 2.02539073775853282867438701364, 3.16895383082278814560872063834, 3.95458717929682004347683026011, 5.60422299514776962173660378239, 6.47484739626348364815222822214, 8.360804839251968326181692232050, 9.325164062387988299191252774941, 10.36999699699015567949446502633, 10.94203814620548455833670326446, 11.50335268868289187366308706635

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