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Properties

Label 2-273-273.242-c1-0-30
Degree $2$
Conductor $273$
Sign $-0.407 + 0.913i$
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  + (1.64 + 0.439i)2-s + (−1.72 − 0.184i)3-s + (0.769 + 0.444i)4-s + (−3.49 − 0.936i)5-s + (−2.74 − 1.06i)6-s + (0.612 − 2.57i)7-s + (−1.33 − 1.33i)8-s + (2.93 + 0.636i)9-s + (−5.32 − 3.07i)10-s + (−0.378 + 0.101i)11-s + (−1.24 − 0.907i)12-s + (−0.988 − 3.46i)13-s + (2.13 − 3.95i)14-s + (5.84 + 2.25i)15-s + (−2.49 − 4.31i)16-s + (−0.773 + 1.33i)17-s + ⋯
L(s)  = 1  + (1.16 + 0.311i)2-s + (−0.994 − 0.106i)3-s + (0.384 + 0.222i)4-s + (−1.56 − 0.419i)5-s + (−1.12 − 0.433i)6-s + (0.231 − 0.972i)7-s + (−0.472 − 0.472i)8-s + (0.977 + 0.212i)9-s + (−1.68 − 0.972i)10-s + (−0.114 + 0.0305i)11-s + (−0.359 − 0.262i)12-s + (−0.274 − 0.961i)13-s + (0.571 − 1.05i)14-s + (1.51 + 0.583i)15-s + (−0.623 − 1.07i)16-s + (−0.187 + 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.407 + 0.913i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438805 - 0.676612i\)
\(L(\frac12)\) \(\approx\) \(0.438805 - 0.676612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.184i)T \)
7 \( 1 + (-0.612 + 2.57i)T \)
13 \( 1 + (0.988 + 3.46i)T \)
good2 \( 1 + (-1.64 - 0.439i)T + (1.73 + i)T^{2} \)
5 \( 1 + (3.49 + 0.936i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.378 - 0.101i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.773 - 1.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.50 - 5.61i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.29iT - 29T^{2} \)
31 \( 1 + (-7.37 + 1.97i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.893 + 0.239i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.40 + 5.40i)T - 41iT^{2} \)
43 \( 1 + 2.78iT - 43T^{2} \)
47 \( 1 + (-1.72 + 6.45i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.185 + 0.107i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.75 + 2.07i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.29 + 9.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.139 - 0.0374i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.99 + 2.99i)T - 71iT^{2} \)
73 \( 1 + (-1.27 - 4.77i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.10 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.55 + 5.55i)T - 83iT^{2} \)
89 \( 1 + (3.50 - 13.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.33 + 6.33i)T + 97iT^{2} \)
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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

−12.06436279055332555489874298816, −10.87758186011846334150231706047, −10.14280532069980380671819273157, −8.277712045835452286804803055768, −7.45949518617521910709236715824, −6.45394329210910374744335914610, −5.25197471797972082276877390735, −4.35451875947877567642079702435, −3.72204074280002725002362568336, −0.48237210011452157280610859374, 2.73246748449428522463844410288, 4.18903634469531974628256201459, 4.72167200118165643585668843266, 5.95933898787031924828865931797, 7.00726393467766144638320872951, 8.218755773793356574151466678756, 9.443078696830167283325340378041, 11.06032035497729176210834909046, 11.61924064525224481879351765125, 11.90985172408848906534090781965

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