Properties

Label 2-273-273.185-c1-0-31
Degree $2$
Conductor $273$
Sign $-0.941 + 0.335i$
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  − 1.73i·3-s + (−1 − 1.73i)4-s + (−2.5 − 0.866i)7-s − 2.99·9-s + (−2.99 + 1.73i)12-s + (1 + 3.46i)13-s + (−1.99 + 3.46i)16-s − 8.66i·19-s + (−1.49 + 4.33i)21-s + (2.5 − 4.33i)25-s + 5.19i·27-s + (1.00 + 5.19i)28-s + (−7.5 − 4.33i)31-s + (2.99 + 5.19i)36-s + (5.5 − 9.52i)37-s + ⋯
L(s)  = 1  − 0.999i·3-s + (−0.5 − 0.866i)4-s + (−0.944 − 0.327i)7-s − 0.999·9-s + (−0.866 + 0.499i)12-s + (0.277 + 0.960i)13-s + (−0.499 + 0.866i)16-s − 1.98i·19-s + (−0.327 + 0.944i)21-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (0.188 + 0.981i)28-s + (−1.34 − 0.777i)31-s + (0.499 + 0.866i)36-s + (0.904 − 1.56i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.941 + 0.335i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.941 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129743 - 0.750152i\)
\(L(\frac12)\) \(\approx\) \(0.129743 - 0.750152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
7 \( 1 + (2.5 + 0.866i)T \)
13 \( 1 + (-1 - 3.46i)T \)
good2 \( 1 + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 8.66iT - 19T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.5 + 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 15.5iT - 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (12 + 6.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.5 - 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.42376862483112469163445446794, −10.65370307297799945834805003725, −9.311893064436845493695050786215, −8.886356997181741808525570788830, −7.27174023488180175545848110315, −6.55919735561680489925539031100, −5.60374273128433385286993151283, −4.16755189048506287075935004495, −2.38166414081353019223230660418, −0.58208706017827288621124243378, 3.10498797724630801994885533464, 3.71976796117377401600108823055, 5.11600618082989964288531664496, 6.15648242665337708955878744780, 7.74223007997395040480041342616, 8.604247727463603858204972038708, 9.514507975512239538876784958308, 10.23319924006779617837292489113, 11.35762938656492406484639141040, 12.47108013941927306407568050761

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