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Properties

Label 2-273-273.2-c1-0-10
Degree $2$
Conductor $273$
Sign $-0.451 - 0.892i$
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  + (1.32 + 1.32i)2-s + (1.33 + 1.10i)3-s + 1.53i·4-s + (−3.58 + 0.961i)5-s + (0.311 + 3.24i)6-s + (0.951 + 2.46i)7-s + (0.617 − 0.617i)8-s + (0.571 + 2.94i)9-s + (−6.04 − 3.49i)10-s + (−1.45 − 5.44i)11-s + (−1.69 + 2.05i)12-s + (3.45 + 1.04i)13-s + (−2.01 + 4.54i)14-s + (−5.85 − 2.66i)15-s + 4.71·16-s + 0.604·17-s + ⋯
L(s)  = 1  + (0.940 + 0.940i)2-s + (0.771 + 0.636i)3-s + 0.767i·4-s + (−1.60 + 0.429i)5-s + (0.127 + 1.32i)6-s + (0.359 + 0.933i)7-s + (0.218 − 0.218i)8-s + (0.190 + 0.981i)9-s + (−1.91 − 1.10i)10-s + (−0.439 − 1.64i)11-s + (−0.488 + 0.592i)12-s + (0.956 + 0.290i)13-s + (−0.538 + 1.21i)14-s + (−1.51 − 0.689i)15-s + 1.17·16-s + 0.146·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.451 - 0.892i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11582 + 1.81565i\)
\(L(\frac12)\) \(\approx\) \(1.11582 + 1.81565i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.33 - 1.10i)T \)
7 \( 1 + (-0.951 - 2.46i)T \)
13 \( 1 + (-3.45 - 1.04i)T \)
good2 \( 1 + (-1.32 - 1.32i)T + 2iT^{2} \)
5 \( 1 + (3.58 - 0.961i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.45 + 5.44i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.604T + 17T^{2} \)
19 \( 1 + (2.61 + 0.700i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 2.73T + 23T^{2} \)
29 \( 1 + (0.0469 - 0.0271i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.75 - 1.27i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (5.11 + 5.11i)T + 37iT^{2} \)
41 \( 1 + (-1.68 + 6.30i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.42 + 2.55i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.552 + 2.06i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.33 - 0.770i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.35 - 7.35i)T - 59iT^{2} \)
61 \( 1 + (0.545 + 0.944i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.27 + 4.75i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.21 + 0.594i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.02 + 3.84i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.56 + 2.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.54 + 5.54i)T + 83iT^{2} \)
89 \( 1 + (9.63 - 9.63i)T - 89iT^{2} \)
97 \( 1 + (2.29 + 8.58i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.34782747568501894733100148573, −11.21605165313558036816523147427, −10.63990024436439667722387167548, −8.743104999829756433073824260329, −8.338831599592736662039181212436, −7.37579953150040420353788713708, −6.09204281406040716940088785954, −4.98047995508200055549358223369, −3.87520796339109317399365160342, −3.12660968445450035128850420941, 1.43159124651419667659978137737, 3.11578566075871821685593820965, 4.10644639781765300453915038701, 4.72022487309436518580801731552, 6.89934983885978673943733980870, 7.85917226295942850937225190386, 8.312708411727141133329998803852, 9.981698911237592110868199235933, 11.05130618575990987627887545466, 11.83397744525097024122920763003

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