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Properties

Label 2-48-48.35-c3-0-13
Degree $2$
Conductor $48$
Sign $-0.999 - 0.0277i$
Analytic cond. $2.83209$
Root an. cond. $1.68288$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 − 1.41i)2-s + (0.563 + 5.16i)3-s + (3.98 + 6.93i)4-s + (−13.1 − 13.1i)5-s + (5.94 − 13.4i)6-s − 13.2·7-s + (0.0933 − 22.6i)8-s + (−26.3 + 5.82i)9-s + (13.5 + 50.8i)10-s + (−24.0 + 24.0i)11-s + (−33.6 + 24.4i)12-s + (−30.7 − 30.7i)13-s + (32.4 + 18.8i)14-s + (60.5 − 75.4i)15-s + (−32.3 + 55.2i)16-s + 56.8i·17-s + ⋯
L(s)  = 1  + (−0.865 − 0.501i)2-s + (0.108 + 0.994i)3-s + (0.497 + 0.867i)4-s + (−1.17 − 1.17i)5-s + (0.404 − 0.914i)6-s − 0.716·7-s + (0.00412 − 0.999i)8-s + (−0.976 + 0.215i)9-s + (0.428 + 1.60i)10-s + (−0.660 + 0.660i)11-s + (−0.808 + 0.588i)12-s + (−0.656 − 0.656i)13-s + (0.620 + 0.359i)14-s + (1.04 − 1.29i)15-s + (−0.504 + 0.863i)16-s + 0.811i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0277i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.999 - 0.0277i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.999 - 0.0277i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.000125407 + 0.00904316i\)
\(L(\frac12)\) \(\approx\) \(0.000125407 + 0.00904316i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.44 + 1.41i)T \)
3 \( 1 + (-0.563 - 5.16i)T \)
good5 \( 1 + (13.1 + 13.1i)T + 125iT^{2} \)
7 \( 1 + 13.2T + 343T^{2} \)
11 \( 1 + (24.0 - 24.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (30.7 + 30.7i)T + 2.19e3iT^{2} \)
17 \( 1 - 56.8iT - 4.91e3T^{2} \)
19 \( 1 + (-74.2 + 74.2i)T - 6.85e3iT^{2} \)
23 \( 1 + 21.7iT - 1.21e4T^{2} \)
29 \( 1 + (102. - 102. i)T - 2.43e4iT^{2} \)
31 \( 1 - 219. iT - 2.97e4T^{2} \)
37 \( 1 + (-83.3 + 83.3i)T - 5.06e4iT^{2} \)
41 \( 1 - 7.92T + 6.89e4T^{2} \)
43 \( 1 + (153. + 153. i)T + 7.95e4iT^{2} \)
47 \( 1 - 208.T + 1.03e5T^{2} \)
53 \( 1 + (390. + 390. i)T + 1.48e5iT^{2} \)
59 \( 1 + (221. - 221. i)T - 2.05e5iT^{2} \)
61 \( 1 + (416. + 416. i)T + 2.26e5iT^{2} \)
67 \( 1 + (284. - 284. i)T - 3.00e5iT^{2} \)
71 \( 1 - 26.9iT - 3.57e5T^{2} \)
73 \( 1 + 839. iT - 3.89e5T^{2} \)
79 \( 1 - 556. iT - 4.93e5T^{2} \)
83 \( 1 + (400. + 400. i)T + 5.71e5iT^{2} \)
89 \( 1 - 974.T + 7.04e5T^{2} \)
97 \( 1 + 1.55e3T + 9.12e5T^{2} \)
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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

−15.23555358619099088759804780010, −12.92085558703566236846471093921, −12.14234631526103184273508116696, −10.82652803311541452748379071904, −9.678485297122131244329611583776, −8.678955585898318963939945302156, −7.55779895582124251379567276785, −4.85869855612152143508399947280, −3.33322561912673590549862993494, −0.008539474063532590146248784115, 2.88885432421623040393097695639, 6.10869737177131954906443429308, 7.30525791670548236374515923775, 7.896055646024863706214154410583, 9.610149059705953031088688299484, 11.13859924036810958238403359875, 11.92069734067773251764110036616, 13.73130547163877047817518187925, 14.74805764230277041574773842470, 15.81563569356512695894282163369

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