Properties

Label 2-273-91.33-c1-0-7
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: $\chi_{273} (124, \cdot )$
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

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

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