Properties

Label 2-273-273.68-c1-0-3
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: $\chi_{273} (68, \cdot )$
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

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

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