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Properties

Label 2-273-273.68-c1-0-23
Degree $2$
Conductor $273$
Sign $-0.0803 + 0.996i$
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  − 0.843i·2-s + (0.346 − 1.69i)3-s + 1.28·4-s + (−1.06 + 1.84i)5-s + (−1.43 − 0.292i)6-s + (2.34 − 1.22i)7-s − 2.77i·8-s + (−2.75 − 1.17i)9-s + (1.55 + 0.896i)10-s + (−0.855 − 0.493i)11-s + (0.446 − 2.18i)12-s + (3.08 − 1.86i)13-s + (−1.03 − 1.97i)14-s + (2.75 + 2.44i)15-s + 0.240·16-s + 1.84·17-s + ⋯
L(s)  = 1  − 0.596i·2-s + (0.200 − 0.979i)3-s + 0.644·4-s + (−0.475 + 0.823i)5-s + (−0.584 − 0.119i)6-s + (0.886 − 0.463i)7-s − 0.980i·8-s + (−0.919 − 0.392i)9-s + (0.490 + 0.283i)10-s + (−0.257 − 0.148i)11-s + (0.129 − 0.631i)12-s + (0.855 − 0.518i)13-s + (−0.276 − 0.528i)14-s + (0.711 + 0.630i)15-s + 0.0600·16-s + 0.446·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0803 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0803 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0803 + 0.996i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0803 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05336 - 1.14175i\)
\(L(\frac12)\) \(\approx\) \(1.05336 - 1.14175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.346 + 1.69i)T \)
7 \( 1 + (-2.34 + 1.22i)T \)
13 \( 1 + (-3.08 + 1.86i)T \)
good2 \( 1 + 0.843iT - 2T^{2} \)
5 \( 1 + (1.06 - 1.84i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.855 + 0.493i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.84T + 17T^{2} \)
19 \( 1 + (6.19 - 3.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.62iT - 23T^{2} \)
29 \( 1 + (0.488 - 0.281i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.23 - 2.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.29T + 37T^{2} \)
41 \( 1 + (-4.93 - 8.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.214 - 0.370i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.40 - 11.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.59 + 3.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 + (-1.92 + 1.10i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.22 + 2.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.48 + 1.43i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.31 - 4.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.27 - 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 - 5.18T + 89T^{2} \)
97 \( 1 + (0.172 + 0.0997i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.34912179544796310650289590215, −11.10760697276233749136880034761, −10.18794915570647663534100257452, −8.420362767354224339923804896437, −7.72908554467228190578254732447, −6.84740288003977919487957606378, −5.89675198715542208647352384166, −3.84880197865358594767421434498, −2.74966585817273650082612277495, −1.39208977000364171642218307899, 2.23668206887387530619840305222, 4.03495578637478341145714396687, 5.04629789046042172439377392620, 5.94880044191817873752971477962, 7.43214293544291330451929893909, 8.570401736020506186783168246149, 8.774683529439775414521974388130, 10.38828981059949290317288798162, 11.27487508438372300633495274059, 11.81850636739594109919790706608

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