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Properties

Label 2-273-21.20-c1-0-28
Degree $2$
Conductor $273$
Sign $-0.943 + 0.331i$
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  − 1.86i·2-s + (0.150 − 1.72i)3-s − 1.46·4-s + 2.96·5-s + (−3.21 − 0.279i)6-s + (−2.41 + 1.09i)7-s − 0.988i·8-s + (−2.95 − 0.518i)9-s − 5.52i·10-s − 1.31i·11-s + (−0.220 + 2.53i)12-s − i·13-s + (2.03 + 4.48i)14-s + (0.445 − 5.11i)15-s − 4.78·16-s + 5.54·17-s + ⋯
L(s)  = 1  − 1.31i·2-s + (0.0867 − 0.996i)3-s − 0.734·4-s + 1.32·5-s + (−1.31 − 0.114i)6-s + (−0.911 + 0.412i)7-s − 0.349i·8-s + (−0.984 − 0.172i)9-s − 1.74i·10-s − 0.396i·11-s + (−0.0637 + 0.731i)12-s − 0.277i·13-s + (0.542 + 1.19i)14-s + (0.115 − 1.32i)15-s − 1.19·16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.943 + 0.331i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.943 + 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.244582 - 1.43405i\)
\(L(\frac12)\) \(\approx\) \(0.244582 - 1.43405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.150 + 1.72i)T \)
7 \( 1 + (2.41 - 1.09i)T \)
13 \( 1 + iT \)
good2 \( 1 + 1.86iT - 2T^{2} \)
5 \( 1 - 2.96T + 5T^{2} \)
11 \( 1 + 1.31iT - 11T^{2} \)
17 \( 1 - 5.54T + 17T^{2} \)
19 \( 1 - 0.857iT - 19T^{2} \)
23 \( 1 - 5.78iT - 23T^{2} \)
29 \( 1 + 3.98iT - 29T^{2} \)
31 \( 1 - 4.64iT - 31T^{2} \)
37 \( 1 - 7.04T + 37T^{2} \)
41 \( 1 - 2.19T + 41T^{2} \)
43 \( 1 - 5.76T + 43T^{2} \)
47 \( 1 + 1.48T + 47T^{2} \)
53 \( 1 - 0.966iT - 53T^{2} \)
59 \( 1 - 4.53T + 59T^{2} \)
61 \( 1 - 7.77iT - 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 + 6.33iT - 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + 0.120iT - 97T^{2} \)
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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

−11.72555458191571374296431373156, −10.52415735110047015874841207348, −9.708092383773273003440049035263, −9.052894901044199732474543113016, −7.56857738073308842344790731631, −6.25444306859003557684994176628, −5.63754719124446374009138191959, −3.35276260982729683419910002961, −2.49163760082820625661399398013, −1.24243792590348355068681289197, 2.69039101145383337974087014489, 4.37684120253927960951116955492, 5.58639163178261523822782486295, 6.16902216807721593450634443157, 7.25169118949907569838857597105, 8.528841360520256589159245941041, 9.555249552848262336515423870017, 9.988420240484095971870192759591, 11.07809714158510443092454718453, 12.58946863273520130274210394995

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