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Properties

Label 2-273-91.4-c1-0-6
Degree $2$
Conductor $273$
Sign $-0.190 - 0.981i$
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  + 2.18i·2-s + (−0.5 − 0.866i)3-s − 2.79·4-s + (1.5 − 0.866i)5-s + (1.89 − 1.09i)6-s + (2.29 + 1.32i)7-s − 1.73i·8-s + (−0.499 + 0.866i)9-s + (1.89 + 3.28i)10-s + (3 − 1.73i)11-s + (1.39 + 2.41i)12-s + (−1 + 3.46i)13-s + (−2.89 + 5.01i)14-s + (−1.5 − 0.866i)15-s − 1.79·16-s + 17-s + ⋯
L(s)  = 1  + 1.54i·2-s + (−0.288 − 0.499i)3-s − 1.39·4-s + (0.670 − 0.387i)5-s + (0.773 − 0.446i)6-s + (0.866 + 0.499i)7-s − 0.612i·8-s + (−0.166 + 0.288i)9-s + (0.599 + 1.03i)10-s + (0.904 − 0.522i)11-s + (0.402 + 0.697i)12-s + (−0.277 + 0.960i)13-s + (−0.773 + 1.34i)14-s + (−0.387 − 0.223i)15-s − 0.447·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.190 - 0.981i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.856389 + 1.03891i\)
\(L(\frac12)\) \(\approx\) \(0.856389 + 1.03891i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.29 - 1.32i)T \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 - 2.18iT - 2T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 + (-4.58 - 2.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.58T + 23T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.29 - 3.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.02iT - 37T^{2} \)
41 \( 1 + (3.08 + 1.77i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.29 + 3.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.708 - 0.409i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.08 + 5.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.28iT - 59T^{2} \)
61 \( 1 + (-2.58 + 4.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.1 - 7.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.87 - 2.23i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.708 - 1.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + 15.5iT - 89T^{2} \)
97 \( 1 + (-9.08 + 5.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.99536359295265684483730653403, −11.66376689500565711884781646254, −9.912127258030561840379595741168, −8.901133125823709047214101167390, −8.155310854777589155121934069361, −7.19095565583651245279357358550, −6.06126600129002490748495954586, −5.59108342635783456532698815034, −4.37516699045673328076898474516, −1.80464224658450772115000481457, 1.34240877351784669010034578133, 2.80985969668751207015783675653, 4.09236626099804177488325026823, 5.07758507948274582718313293962, 6.53140877512309993709980259273, 7.993754811812645998905038465545, 9.364949525855615504629574911458, 10.15754269388081550539956313172, 10.50997557466781758900427363355, 11.79263172664726810954293366618

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