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Properties

Label 2-273-91.25-c1-0-8
Degree $2$
Conductor $273$
Sign $0.879 + 0.475i$
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 + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−3 + 1.73i)5-s + 1.73i·6-s + (0.5 − 2.59i)7-s − 1.73i·8-s + (−0.499 − 0.866i)9-s + (3 − 5.19i)10-s + (4.5 + 2.59i)11-s + (−0.500 − 0.866i)12-s + (−1 − 3.46i)13-s + (1.5 + 4.33i)14-s + 3.46i·15-s + (2.49 + 4.33i)16-s + (1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 − 0.433i)4-s + (−1.34 + 0.774i)5-s + 0.707i·6-s + (0.188 − 0.981i)7-s − 0.612i·8-s + (−0.166 − 0.288i)9-s + (0.948 − 1.64i)10-s + (1.35 + 0.783i)11-s + (−0.144 − 0.249i)12-s + (−0.277 − 0.960i)13-s + (0.400 + 1.15i)14-s + 0.894i·15-s + (0.624 + 1.08i)16-s + (0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593632 - 0.150335i\)
\(L(\frac12)\) \(\approx\) \(0.593632 - 0.150335i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (1 + 3.46i)T \)
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 \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 1.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (1.5 - 0.866i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.5 + 4.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + (9 + 5.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (-3 + 1.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.3iT - 97T^{2} \)
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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

−11.84440254993144107741731882980, −10.67513614046649055393601100204, −9.819760625622198764040560566659, −8.683858897455107663618442120565, −7.65497830786548578635213910945, −7.31011543516290121031769716270, −6.58677003388973727867381772807, −4.33954555862275919588376019151, −3.27596609173064951824024846724, −0.74559322105855234028139010698, 1.43928483439780298724367027967, 3.35632158410707041133788598801, 4.51359882383000845563905513325, 5.85173327134042697677499112049, 7.73894943230098773934158074681, 8.445724025336055315310301250677, 9.121872411717637675563979785834, 9.761749774580147640585612782666, 11.23718717750221031381516919891, 11.76371530990473497010391220833

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