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Properties

Label 2-273-91.30-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.0383 + 0.999i$
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  + (2.40 − 1.39i)2-s − 3-s + (2.86 − 4.96i)4-s + (0.804 + 0.464i)5-s + (−2.40 + 1.39i)6-s + (−2.63 + 0.283i)7-s − 10.3i·8-s + 9-s + 2.58·10-s + 1.94i·11-s + (−2.86 + 4.96i)12-s + (2.64 + 2.44i)13-s + (−5.94 + 4.34i)14-s + (−0.804 − 0.464i)15-s + (−8.70 − 15.0i)16-s + (0.403 − 0.698i)17-s + ⋯
L(s)  = 1  + (1.70 − 0.983i)2-s − 0.577·3-s + (1.43 − 2.48i)4-s + (0.359 + 0.207i)5-s + (−0.983 + 0.567i)6-s + (−0.994 + 0.107i)7-s − 3.67i·8-s + 0.333·9-s + 0.816·10-s + 0.586i·11-s + (−0.827 + 1.43i)12-s + (0.734 + 0.678i)13-s + (−1.58 + 1.16i)14-s + (−0.207 − 0.119i)15-s + (−2.17 − 3.77i)16-s + (0.0978 − 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0383 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0383 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0383 + 0.999i$
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: \((2,\ 273,\ (\ :1/2),\ -0.0383 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82020 - 1.89138i\)
\(L(\frac12)\) \(\approx\) \(1.82020 - 1.89138i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.283i)T \)
13 \( 1 + (-2.64 - 2.44i)T \)
good2 \( 1 + (-2.40 + 1.39i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.804 - 0.464i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.94iT - 11T^{2} \)
17 \( 1 + (-0.403 + 0.698i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 3.15iT - 19T^{2} \)
23 \( 1 + (-2.87 - 4.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.963 - 1.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.84 - 2.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.41 + 2.54i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.68 + 0.972i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.65 - 9.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (11.1 + 6.45i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.22 + 5.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.14 + 0.659i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.90T + 61T^{2} \)
67 \( 1 + 0.206iT - 67T^{2} \)
71 \( 1 + (-1.89 + 1.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.49 - 4.90i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.58 - 7.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.6iT - 83T^{2} \)
89 \( 1 + (2.74 - 1.58i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.3 + 5.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.76807409058612596322062970413, −11.09523229523177975511693954847, −10.07254723539638990677755485485, −9.502729966513814964556021319491, −7.02388541909302436756377716490, −6.23672826997001054500615202018, −5.46917080638872157117981329324, −4.22798770905595032807856074496, −3.21027327209413419161494726008, −1.72547472587820717138214574333, 2.91898977664743773841037045017, 3.99667276185834142758691308071, 5.26441228462381575949184463651, 6.02872326173018513767189909957, 6.69484089917227409748812048257, 7.81152784452329178173550123170, 9.040173928474795492588260346997, 10.69475069181443221251930600611, 11.51742245111381576022573601755, 12.65206815465630170235653248881

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