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Properties

Label 2-273-91.30-c1-0-14
Degree $2$
Conductor $273$
Sign $0.923 + 0.384i$
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  + (1.5 − 0.866i)2-s − 3-s + (0.5 − 0.866i)4-s + (3 + 1.73i)5-s + (−1.5 + 0.866i)6-s + (0.5 − 2.59i)7-s + 1.73i·8-s + 9-s + 6·10-s + 3.46i·11-s + (−0.5 + 0.866i)12-s + (−2.5 − 2.59i)13-s + (−1.5 − 4.33i)14-s + (−3 − 1.73i)15-s + (2.49 + 4.33i)16-s + (3 − 5.19i)17-s + ⋯
L(s)  = 1  + (1.06 − 0.612i)2-s − 0.577·3-s + (0.250 − 0.433i)4-s + (1.34 + 0.774i)5-s + (−0.612 + 0.353i)6-s + (0.188 − 0.981i)7-s + 0.612i·8-s + 0.333·9-s + 1.89·10-s + 1.04i·11-s + (−0.144 + 0.250i)12-s + (−0.693 − 0.720i)13-s + (−0.400 − 1.15i)14-s + (−0.774 − 0.447i)15-s + (0.624 + 1.08i)16-s + (0.727 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.923 + 0.384i$
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: \((2,\ 273,\ (\ :1/2),\ 0.923 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07342 - 0.414869i\)
\(L(\frac12)\) \(\approx\) \(2.07342 - 0.414869i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-3 - 1.73i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (9 - 5.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3 - 1.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (9 + 5.19i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + (-3 + 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.5 + 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (-9 + 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.5 - 6.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.02680280425137614838726726855, −10.82745687674033883816105492775, −10.36704638556502135354665908695, −9.475193063919237891932553386998, −7.55101169302945768504384020485, −6.71343202063850321882504079597, −5.39847993891985409024041133159, −4.76794308105046151419677111958, −3.23920554711996788314761607259, −1.98910178341815102602511727387, 1.82197219904383170117892809652, 3.89579889744743059537468720891, 5.35962887463810102896481840051, 5.64953569134817171962475453745, 6.34523879151231982046717751988, 7.945763775133977894826426782610, 9.306919678088894343197467606899, 9.842863745000419376528202222235, 11.29652471530466281221861079199, 12.36182992480258126930711934657

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