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Properties

Label 2-273-7.2-c1-0-8
Degree $2$
Conductor $273$
Sign $0.595 + 0.803i$
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  + (−1.13 + 1.96i)2-s + (−0.5 − 0.866i)3-s + (−1.56 − 2.71i)4-s + (−1.63 + 2.82i)5-s + 2.26·6-s + (−0.133 − 2.64i)7-s + 2.57·8-s + (−0.499 + 0.866i)9-s + (−3.70 − 6.41i)10-s + (−1.06 − 1.84i)11-s + (−1.56 + 2.71i)12-s − 13-s + (5.33 + 2.73i)14-s + 3.26·15-s + (0.219 − 0.379i)16-s + (−1.72 − 2.98i)17-s + ⋯
L(s)  = 1  + (−0.801 + 1.38i)2-s + (−0.288 − 0.499i)3-s + (−0.783 − 1.35i)4-s + (−0.730 + 1.26i)5-s + 0.925·6-s + (−0.0503 − 0.998i)7-s + 0.910·8-s + (−0.166 + 0.288i)9-s + (−1.17 − 2.02i)10-s + (−0.321 − 0.556i)11-s + (−0.452 + 0.783i)12-s − 0.277·13-s + (1.42 + 0.730i)14-s + 0.843·15-s + (0.0547 − 0.0948i)16-s + (−0.417 − 0.723i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.595 + 0.803i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222245 - 0.111984i\)
\(L(\frac12)\) \(\approx\) \(0.222245 - 0.111984i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.133 + 2.64i)T \)
13 \( 1 + T \)
good2 \( 1 + (1.13 - 1.96i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.63 - 2.82i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.06 + 1.84i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.72 + 2.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.635 + 1.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.48 + 7.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.53T + 29T^{2} \)
31 \( 1 + (-1.78 - 3.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.890 + 1.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.95T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + (3.61 - 6.25i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.60 + 9.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.97 + 5.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.45 - 2.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.20 + 5.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.23T + 71T^{2} \)
73 \( 1 + (-0.479 - 0.830i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.65 - 2.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + (7.54 - 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.78T + 97T^{2} \)
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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

−11.35904935295941156733508416947, −10.82083311730738212587234552076, −9.756096180182395241517125809322, −8.448287209978795466600742873290, −7.56560411073891741943486382611, −6.97302770799706710515599340521, −6.39192285417064218496167957287, −4.89431697713483616291320575044, −3.12635483804561805098132667156, −0.25486261541803947807013304473, 1.70716033245301186011850623982, 3.36238558270351181296825270419, 4.57116767157549316935372648647, 5.65517398418649807434044503104, 7.71483054304205957734371754915, 8.692494733470636800472127174726, 9.288411269971388574025159771181, 10.06124456769063276503764982634, 11.36861252618352037223657572415, 11.70187326288099386026740768192

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