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Properties

Label 2-273-273.251-c1-0-8
Degree $2$
Conductor $273$
Sign $0.962 - 0.269i$
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  + (0.835 − 1.44i)2-s + (−1.72 + 0.174i)3-s + (−0.395 − 0.685i)4-s + 2.15i·5-s + (−1.18 + 2.63i)6-s + (−0.338 + 2.62i)7-s + 2.01·8-s + (2.93 − 0.601i)9-s + (3.11 + 1.79i)10-s + (−2.84 + 4.92i)11-s + (0.801 + 1.11i)12-s + (0.842 − 3.50i)13-s + (3.51 + 2.68i)14-s + (−0.375 − 3.70i)15-s + (2.47 − 4.29i)16-s + (2.39 + 4.14i)17-s + ⋯
L(s)  = 1  + (0.590 − 1.02i)2-s + (−0.994 + 0.100i)3-s + (−0.197 − 0.342i)4-s + 0.962i·5-s + (−0.484 + 1.07i)6-s + (−0.128 + 0.991i)7-s + 0.713·8-s + (0.979 − 0.200i)9-s + (0.984 + 0.568i)10-s + (−0.857 + 1.48i)11-s + (0.231 + 0.321i)12-s + (0.233 − 0.972i)13-s + (0.939 + 0.716i)14-s + (−0.0969 − 0.957i)15-s + (0.619 − 1.07i)16-s + (0.579 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.962 - 0.269i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.962 - 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27467 + 0.175086i\)
\(L(\frac12)\) \(\approx\) \(1.27467 + 0.175086i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.174i)T \)
7 \( 1 + (0.338 - 2.62i)T \)
13 \( 1 + (-0.842 + 3.50i)T \)
good2 \( 1 + (-0.835 + 1.44i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.15iT - 5T^{2} \)
11 \( 1 + (2.84 - 4.92i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.39 - 4.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.84 + 3.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.16 - 1.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.76 + 1.01i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.53T + 31T^{2} \)
37 \( 1 + (5.72 + 3.30i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.25 - 0.722i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.58 - 7.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.512iT - 47T^{2} \)
53 \( 1 + 3.84iT - 53T^{2} \)
59 \( 1 + (-4.63 + 2.67i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.3 + 5.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.01 + 4.05i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.01 + 8.68i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.06T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 1.99iT - 83T^{2} \)
89 \( 1 + (-4.43 - 2.55i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.539 + 0.935i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.00426265086034793981239402438, −10.96433621736063646011477336416, −10.51655004171856547767551130388, −9.691280467642512555226211809326, −7.86430453278641331778522369371, −6.86269559873780632456097707970, −5.62582965232452459280678649503, −4.69309087425068575474591214216, −3.25826813445038386596445775500, −2.09477760739244713832252025370, 1.00445524186531869758623856602, 3.98701574514413965851759842911, 5.01253144001942248057445939748, 5.71402472301926866577703197245, 6.74690212651520973373194234178, 7.60156564953004272560644446267, 8.721004423991359923100835074133, 10.22480087886027042316481211140, 10.91980802961023179920252125417, 11.96925022635431512282627814127

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