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Properties

Label 2-273-91.45-c1-0-10
Degree $2$
Conductor $273$
Sign $0.943 + 0.332i$
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  + (−0.184 + 0.184i)2-s + (−0.866 + 0.5i)3-s + 1.93i·4-s + (0.900 − 3.35i)5-s + (0.0673 − 0.251i)6-s + (0.939 − 2.47i)7-s + (−0.723 − 0.723i)8-s + (0.499 − 0.866i)9-s + (0.452 + 0.784i)10-s + (0.621 − 2.31i)11-s + (−0.966 − 1.67i)12-s + (1.60 + 3.23i)13-s + (0.282 + 0.628i)14-s + (0.900 + 3.35i)15-s − 3.59·16-s + 3.34·17-s + ⋯
L(s)  = 1  + (−0.130 + 0.130i)2-s + (−0.499 + 0.288i)3-s + 0.966i·4-s + (0.402 − 1.50i)5-s + (0.0275 − 0.102i)6-s + (0.355 − 0.934i)7-s + (−0.255 − 0.255i)8-s + (0.166 − 0.288i)9-s + (0.143 + 0.247i)10-s + (0.187 − 0.699i)11-s + (−0.278 − 0.483i)12-s + (0.443 + 0.896i)13-s + (0.0754 + 0.167i)14-s + (0.232 + 0.867i)15-s − 0.899·16-s + 0.810·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.943 + 0.332i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.943 + 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11659 - 0.190902i\)
\(L(\frac12)\) \(\approx\) \(1.11659 - 0.190902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.939 + 2.47i)T \)
13 \( 1 + (-1.60 - 3.23i)T \)
good2 \( 1 + (0.184 - 0.184i)T - 2iT^{2} \)
5 \( 1 + (-0.900 + 3.35i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.621 + 2.31i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 3.34T + 17T^{2} \)
19 \( 1 + (-6.06 + 1.62i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.157iT - 23T^{2} \)
29 \( 1 + (-0.530 + 0.919i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.14 - 1.37i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.73 + 1.73i)T + 37iT^{2} \)
41 \( 1 + (-11.0 + 2.95i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.74 - 3.89i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.60 + 2.03i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.19 - 5.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.54 - 2.54i)T - 59iT^{2} \)
61 \( 1 + (-10.8 - 6.24i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.12 + 1.10i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.09 - 0.829i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.97 - 7.36i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.17 - 5.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.4 + 10.4i)T + 83iT^{2} \)
89 \( 1 + (10.1 - 10.1i)T - 89iT^{2} \)
97 \( 1 + (-1.40 + 5.24i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.81876293666222328530744325155, −11.16214004442566304602298271761, −9.712214513532698214279726515462, −8.991651709662140649023444051279, −8.073050815062081295498626255057, −7.04353073761006560635146809725, −5.64533362025442676570253229616, −4.59153004226418047367082116438, −3.62007132419139238613987262452, −1.13161976464362083863816301562, 1.72861415946057064980432245562, 3.06990022026645824920717759881, 5.24946413558737575842514223348, 5.88138731026475609450241413777, 6.82728476358971756305143794733, 7.918824590405296999776811737407, 9.519286887798173896176456846267, 10.11251460472674732469773832176, 11.03082855959801856317413385666, 11.64215571122744021299534561265

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