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Properties

Label 2-48-16.13-c5-0-13
Degree $2$
Conductor $48$
Sign $0.998 + 0.0624i$
Analytic cond. $7.69842$
Root an. cond. $2.77460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.58 + 5.02i)2-s + (−6.36 − 6.36i)3-s + (−18.6 + 26.0i)4-s + (74.2 − 74.2i)5-s + (15.5 − 48.4i)6-s − 68.3i·7-s + (−179. − 26.1i)8-s + 81i·9-s + (565. + 181. i)10-s + (404. − 404. i)11-s + (284. − 47.3i)12-s + (714. + 714. i)13-s + (344. − 177. i)14-s − 945.·15-s + (−331. − 968. i)16-s + 171.·17-s + ⋯
L(s)  = 1  + (0.457 + 0.889i)2-s + (−0.408 − 0.408i)3-s + (−0.581 + 0.813i)4-s + (1.32 − 1.32i)5-s + (0.176 − 0.549i)6-s − 0.527i·7-s + (−0.989 − 0.144i)8-s + 0.333i·9-s + (1.78 + 0.573i)10-s + (1.00 − 1.00i)11-s + (0.569 − 0.0948i)12-s + (1.17 + 1.17i)13-s + (0.469 − 0.241i)14-s − 1.08·15-s + (−0.324 − 0.945i)16-s + 0.143·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0624i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.998 + 0.0624i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ 0.998 + 0.0624i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.12308 - 0.0663947i\)
\(L(\frac12)\) \(\approx\) \(2.12308 - 0.0663947i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.58 - 5.02i)T \)
3 \( 1 + (6.36 + 6.36i)T \)
good5 \( 1 + (-74.2 + 74.2i)T - 3.12e3iT^{2} \)
7 \( 1 + 68.3iT - 1.68e4T^{2} \)
11 \( 1 + (-404. + 404. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-714. - 714. i)T + 3.71e5iT^{2} \)
17 \( 1 - 171.T + 1.41e6T^{2} \)
19 \( 1 + (842. + 842. i)T + 2.47e6iT^{2} \)
23 \( 1 - 324. iT - 6.43e6T^{2} \)
29 \( 1 + (-409. - 409. i)T + 2.05e7iT^{2} \)
31 \( 1 + 5.91e3T + 2.86e7T^{2} \)
37 \( 1 + (-3.99e3 + 3.99e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.41e4iT - 1.15e8T^{2} \)
43 \( 1 + (1.14e4 - 1.14e4i)T - 1.47e8iT^{2} \)
47 \( 1 - 6.77e3T + 2.29e8T^{2} \)
53 \( 1 + (1.48e4 - 1.48e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (7.37e3 - 7.37e3i)T - 7.14e8iT^{2} \)
61 \( 1 + (-1.71e3 - 1.71e3i)T + 8.44e8iT^{2} \)
67 \( 1 + (-4.81e3 - 4.81e3i)T + 1.35e9iT^{2} \)
71 \( 1 - 6.22e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.14e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.16e3T + 3.07e9T^{2} \)
83 \( 1 + (-1.37e4 - 1.37e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 5.09e4iT - 5.58e9T^{2} \)
97 \( 1 - 5.41e4T + 8.58e9T^{2} \)
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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

−14.22777343785418141171013148083, −13.56443311370107570179909512397, −12.81530688767087135482844813706, −11.40425319622002583730969797575, −9.323924881945451177911513766253, −8.535424106116330065803801119298, −6.57926378777762587148963545217, −5.77171623199844940237380802734, −4.29319567848913995192754825730, −1.18981862951514112745283772514, 1.89885941493359369048791204241, 3.50986998016904554659273521675, 5.54666131553329200727408472237, 6.45308140283181438824540346907, 9.190898193291313378950302832676, 10.22798253929534560553443890108, 10.90285291648341945075783712000, 12.27052787796817157621449818451, 13.49102399608246072669410054883, 14.62216876976844036301868910798

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