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Properties

Label 2-48-48.35-c3-0-5
Degree $2$
Conductor $48$
Sign $0.242 - 0.970i$
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 + (−5.16 − 0.563i)3-s + (3.98 + 6.93i)4-s + (13.1 + 13.1i)5-s + (−11.8 − 8.70i)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 + (−16.6 − 38.0i)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 + (−0.994 − 0.108i)3-s + (0.497 + 0.867i)4-s + (1.17 + 1.17i)5-s + (−0.805 − 0.592i)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 + (−0.400 − 0.916i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.242 - 0.970i$
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: \((2,\ 48,\ (\ :3/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.41430 + 1.10414i\)
\(L(\frac12)\) \(\approx\) \(1.41430 + 1.10414i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.16 + 0.563i)T \)
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

−15.40127832885623803322530300329, −14.07220592547578974816856524449, −13.35276929340758508427934810146, −12.02945322916397318729170660761, −10.90335148650892294676038849463, −9.671608461219558005335788097386, −7.13606988842825956953039347709, −6.37037263868638358798760615594, −5.29116845523752135358516845711, −2.97404973827269097712212555463, 1.52110453444286401239134140330, 4.38598742539127202804887703206, 5.59078327073439414780293152790, 6.61161724814961357689977693713, 9.541323574339827701937756214519, 10.05936999562042960196534452663, 11.82986154731270806599662107726, 12.57185577903592939574002402176, 13.39185202908982690029209883197, 14.74805161059845795982663842134

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