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Properties

Label 2-48-48.11-c3-0-17
Degree $2$
Conductor $48$
Sign $0.987 + 0.155i$
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  + (2.81 + 0.221i)2-s + (4.43 − 2.71i)3-s + (7.90 + 1.24i)4-s + (−3.17 + 3.17i)5-s + (13.0 − 6.66i)6-s − 32.3·7-s + (22.0 + 5.26i)8-s + (12.2 − 24.0i)9-s + (−9.65 + 8.25i)10-s + (16.0 + 16.0i)11-s + (38.4 − 15.9i)12-s + (−18.2 + 18.2i)13-s + (−91.2 − 7.15i)14-s + (−5.45 + 22.6i)15-s + (60.8 + 19.7i)16-s + 38.5i·17-s + ⋯
L(s)  = 1  + (0.996 + 0.0781i)2-s + (0.852 − 0.522i)3-s + (0.987 + 0.155i)4-s + (−0.284 + 0.284i)5-s + (0.891 − 0.453i)6-s − 1.74·7-s + (0.972 + 0.232i)8-s + (0.454 − 0.890i)9-s + (−0.305 + 0.260i)10-s + (0.441 + 0.441i)11-s + (0.923 − 0.382i)12-s + (−0.390 + 0.390i)13-s + (−1.74 − 0.136i)14-s + (−0.0939 + 0.390i)15-s + (0.951 + 0.307i)16-s + 0.550i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.987 + 0.155i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ 0.987 + 0.155i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.53262 - 0.198429i\)
\(L(\frac12)\) \(\approx\) \(2.53262 - 0.198429i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.81 - 0.221i)T \)
3 \( 1 + (-4.43 + 2.71i)T \)
good5 \( 1 + (3.17 - 3.17i)T - 125iT^{2} \)
7 \( 1 + 32.3T + 343T^{2} \)
11 \( 1 + (-16.0 - 16.0i)T + 1.33e3iT^{2} \)
13 \( 1 + (18.2 - 18.2i)T - 2.19e3iT^{2} \)
17 \( 1 - 38.5iT - 4.91e3T^{2} \)
19 \( 1 + (56.2 + 56.2i)T + 6.85e3iT^{2} \)
23 \( 1 + 197. iT - 1.21e4T^{2} \)
29 \( 1 + (-57.3 - 57.3i)T + 2.43e4iT^{2} \)
31 \( 1 - 148. iT - 2.97e4T^{2} \)
37 \( 1 + (72.5 + 72.5i)T + 5.06e4iT^{2} \)
41 \( 1 - 73.1T + 6.89e4T^{2} \)
43 \( 1 + (-226. + 226. i)T - 7.95e4iT^{2} \)
47 \( 1 - 412.T + 1.03e5T^{2} \)
53 \( 1 + (-94.8 + 94.8i)T - 1.48e5iT^{2} \)
59 \( 1 + (344. + 344. i)T + 2.05e5iT^{2} \)
61 \( 1 + (153. - 153. i)T - 2.26e5iT^{2} \)
67 \( 1 + (603. + 603. i)T + 3.00e5iT^{2} \)
71 \( 1 - 711. iT - 3.57e5T^{2} \)
73 \( 1 - 687. iT - 3.89e5T^{2} \)
79 \( 1 + 162. iT - 4.93e5T^{2} \)
83 \( 1 + (748. - 748. i)T - 5.71e5iT^{2} \)
89 \( 1 - 927.T + 7.04e5T^{2} \)
97 \( 1 + 208.T + 9.12e5T^{2} \)
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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

−14.95687011071345819273998689007, −13.97521332443608350195206217820, −12.77588820640861637564636885322, −12.34102262385203398536671546883, −10.45468980977556385274843433124, −8.973401319977575622035699605123, −7.11423344592214372141821972075, −6.46345512068482108409096989333, −3.98215888563093824555945360883, −2.66218937321514688217670389395, 2.94777438260998080822438064008, 4.09173938857913151582182759850, 5.98881202441967398515573671064, 7.56240064457436801561154852723, 9.328709800418920312886152645330, 10.37030280648819261650828507219, 12.03934359547292141081274655668, 13.12195576831891571625417271654, 13.91872861244632337118601870451, 15.23419426395978370252275587525

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