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Properties

Label 2-273-91.30-c1-0-11
Degree $2$
Conductor $273$
Sign $0.997 - 0.0742i$
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  + (0.845 − 0.488i)2-s + 3-s + (−0.523 + 0.906i)4-s + (−0.233 − 0.134i)5-s + (0.845 − 0.488i)6-s + (2.62 + 0.292i)7-s + 2.97i·8-s + 9-s − 0.263·10-s + 0.856i·11-s + (−0.523 + 0.906i)12-s + (1.45 − 3.29i)13-s + (2.36 − 1.03i)14-s + (−0.233 − 0.134i)15-s + (0.406 + 0.703i)16-s + (−1.08 + 1.88i)17-s + ⋯
L(s)  = 1  + (0.598 − 0.345i)2-s + 0.577·3-s + (−0.261 + 0.453i)4-s + (−0.104 − 0.0603i)5-s + (0.345 − 0.199i)6-s + (0.993 + 0.110i)7-s + 1.05i·8-s + 0.333·9-s − 0.0833·10-s + 0.258i·11-s + (−0.151 + 0.261i)12-s + (0.402 − 0.915i)13-s + (0.632 − 0.277i)14-s + (−0.0603 − 0.0348i)15-s + (0.101 + 0.175i)16-s + (−0.263 + 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.997 - 0.0742i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.997 - 0.0742i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98725 + 0.0738578i\)
\(L(\frac12)\) \(\approx\) \(1.98725 + 0.0738578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (-2.62 - 0.292i)T \)
13 \( 1 + (-1.45 + 3.29i)T \)
good2 \( 1 + (-0.845 + 0.488i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.233 + 0.134i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.856iT - 11T^{2} \)
17 \( 1 + (1.08 - 1.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.99iT - 19T^{2} \)
23 \( 1 + (0.961 + 1.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.97 - 5.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.0946 + 0.0546i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.02 - 1.74i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.03 + 4.05i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.78 + 3.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.592 - 0.342i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.21 + 7.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.33 - 2.50i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 5.58iT - 67T^{2} \)
71 \( 1 + (12.5 - 7.22i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.56 + 2.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.782 - 1.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.98iT - 83T^{2} \)
89 \( 1 + (2.35 - 1.35i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.93 - 5.15i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.05806661391518968586380519617, −11.15924676993487399323570685139, −10.17548021147501697690414893344, −8.625710567661153309701504181129, −8.340401878393804091897965117010, −7.20413335385769172186435388633, −5.49109998373508362403631790220, −4.49707563203070424860532698409, −3.44487943466636816318721460362, −2.09575833509670403082589535979, 1.68788105131360941305687756453, 3.69012884254151153215784136560, 4.62545326517535221471964037352, 5.72444398138234579161273306638, 6.91255260856264379983510655227, 7.959840001617680758952888842405, 9.021460425946724167196563042394, 9.870182833581392269014387525749, 11.05317270558794504148658987194, 11.88226358472484716323745001206

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