Properties

Label 2-273-13.3-c1-0-4
Degree $2$
Conductor $273$
Sign $0.5 + 0.866i$
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  + (−0.825 − 1.42i)2-s + (0.5 + 0.866i)3-s + (−0.363 + 0.628i)4-s + 2.92·5-s + (0.825 − 1.42i)6-s + (0.5 − 0.866i)7-s − 2.10·8-s + (−0.499 + 0.866i)9-s + (−2.41 − 4.18i)10-s + (2.18 + 3.79i)11-s − 0.726·12-s + (2.56 − 2.53i)13-s − 1.65·14-s + (1.46 + 2.53i)15-s + (2.46 + 4.26i)16-s + (−1.82 + 3.16i)17-s + ⋯
L(s)  = 1  + (−0.583 − 1.01i)2-s + (0.288 + 0.499i)3-s + (−0.181 + 0.314i)4-s + 1.30·5-s + (0.337 − 0.583i)6-s + (0.188 − 0.327i)7-s − 0.743·8-s + (−0.166 + 0.288i)9-s + (−0.763 − 1.32i)10-s + (0.659 + 1.14i)11-s − 0.209·12-s + (0.711 − 0.702i)13-s − 0.441·14-s + (0.377 + 0.654i)15-s + (0.615 + 1.06i)16-s + (−0.442 + 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13147 - 0.653258i\)
\(L(\frac12)\) \(\approx\) \(1.13147 - 0.653258i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-2.56 + 2.53i)T \)
good2 \( 1 + (0.825 + 1.42i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 2.92T + 5T^{2} \)
11 \( 1 + (-2.18 - 3.79i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.82 - 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.87 + 4.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.51 + 6.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.599 + 1.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.47T + 31T^{2} \)
37 \( 1 + (-1.46 - 2.53i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.30 - 7.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.86 + 4.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.58T + 47T^{2} \)
53 \( 1 + 0.302T + 53T^{2} \)
59 \( 1 + (1.51 - 2.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.151 + 0.261i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.35 - 7.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.76 - 11.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.932T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + 1.26T + 83T^{2} \)
89 \( 1 + (-6.69 - 11.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.96 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.38906204018366641464836680469, −10.59382089331502203647585469352, −9.890110917922334228703878876456, −9.314814715479968904336548063296, −8.369382931361198451205264989819, −6.71019786842470768018950689351, −5.65078213809830347559893574763, −4.19826478399693015389396201591, −2.66517276448428684285057340624, −1.55789646908467410830852969784, 1.72022681880131498204131089045, 3.39658591620786792295248998291, 5.72154763964279330404249440879, 6.04883629938477222855105806303, 7.16006807535301072695437214042, 8.196249518426549168446069696242, 9.167427809697754660256036730837, 9.518483608814491546872244242673, 11.18038891053312906821367580773, 12.02568604074431911876764812079

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