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Properties

Label 2-273-273.269-c1-0-17
Degree $2$
Conductor $273$
Sign $0.929 + 0.370i$
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  + 2.73i·2-s + (−1.52 − 0.816i)3-s − 5.46·4-s + (0.121 + 0.209i)5-s + (2.22 − 4.17i)6-s + (−2.13 − 1.56i)7-s − 9.45i·8-s + (1.66 + 2.49i)9-s + (−0.573 + 0.331i)10-s + (4.30 − 2.48i)11-s + (8.34 + 4.45i)12-s + (−3.55 − 0.578i)13-s + (4.27 − 5.82i)14-s + (−0.0137 − 0.419i)15-s + 14.8·16-s − 3.50·17-s + ⋯
L(s)  = 1  + 1.93i·2-s + (−0.881 − 0.471i)3-s − 2.73·4-s + (0.0542 + 0.0938i)5-s + (0.910 − 1.70i)6-s + (−0.806 − 0.591i)7-s − 3.34i·8-s + (0.555 + 0.831i)9-s + (−0.181 + 0.104i)10-s + (1.29 − 0.750i)11-s + (2.40 + 1.28i)12-s + (−0.987 − 0.160i)13-s + (1.14 − 1.55i)14-s + (−0.00355 − 0.108i)15-s + 3.72·16-s − 0.849·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.929 + 0.370i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.929 + 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.348783 - 0.0669071i\)
\(L(\frac12)\) \(\approx\) \(0.348783 - 0.0669071i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.52 + 0.816i)T \)
7 \( 1 + (2.13 + 1.56i)T \)
13 \( 1 + (3.55 + 0.578i)T \)
good2 \( 1 - 2.73iT - 2T^{2} \)
5 \( 1 + (-0.121 - 0.209i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.30 + 2.48i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.50T + 17T^{2} \)
19 \( 1 + (1.51 + 0.873i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (0.841 + 0.485i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.95 + 2.85i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.98T + 37T^{2} \)
41 \( 1 + (0.800 - 1.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.17 + 2.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0888 - 0.153i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.86 + 2.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.71T + 59T^{2} \)
61 \( 1 + (7.71 + 4.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.643 + 1.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.46 + 2.00i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.79 - 1.61i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.62 - 2.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + 9.74T + 89T^{2} \)
97 \( 1 + (-4.44 + 2.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.22343031232795093512208677221, −10.72913169709064446204572200952, −9.642951479791944986377159545291, −8.702676607286509037780077303248, −7.51513059110997239330712876665, −6.60905355167509036448717830371, −6.34830410194988641585833327541, −5.05246036845848969229383665960, −4.00044528362193315523797726954, −0.31001267889134482392441311437, 1.77599621169503185911003534162, 3.39917136630922231177174849338, 4.41512029096480333162342065697, 5.39374787382794449995156368412, 6.86114046045182723277934226148, 9.069333877507842187974353918632, 9.335022360073826866243332076953, 10.21446262064988078859692389485, 11.11922118193901914004234873148, 12.00241325220061790353445192330

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