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Properties

Label 2-48-48.29-c2-0-7
Degree $2$
Conductor $48$
Sign $0.978 + 0.205i$
Analytic cond. $1.30790$
Root an. cond. $1.14363$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.68 − 1.08i)2-s + (0.737 + 2.90i)3-s + (1.64 − 3.64i)4-s + (−1.57 − 1.57i)5-s + (4.39 + 4.08i)6-s + 3.64i·7-s + (−1.19 − 7.91i)8-s + (−7.91 + 4.29i)9-s + (−4.35 − 0.937i)10-s + (−1.19 − 1.19i)11-s + (11.8 + 2.09i)12-s + (−14.6 − 14.6i)13-s + (3.95 + 6.12i)14-s + (3.41 − 5.74i)15-s + (−10.5 − 12.0i)16-s + 28.0i·17-s + ⋯
L(s)  = 1  + (0.840 − 0.542i)2-s + (0.245 + 0.969i)3-s + (0.411 − 0.911i)4-s + (−0.314 − 0.314i)5-s + (0.732 + 0.680i)6-s + 0.520i·7-s + (−0.148 − 0.988i)8-s + (−0.878 + 0.476i)9-s + (−0.435 − 0.0937i)10-s + (−0.108 − 0.108i)11-s + (0.984 + 0.174i)12-s + (−1.12 − 1.12i)13-s + (0.282 + 0.437i)14-s + (0.227 − 0.382i)15-s + (−0.661 − 0.750i)16-s + 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.978 + 0.205i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ 0.978 + 0.205i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64712 - 0.170954i\)
\(L(\frac12)\) \(\approx\) \(1.64712 - 0.170954i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.68 + 1.08i)T \)
3 \( 1 + (-0.737 - 2.90i)T \)
good5 \( 1 + (1.57 + 1.57i)T + 25iT^{2} \)
7 \( 1 - 3.64iT - 49T^{2} \)
11 \( 1 + (1.19 + 1.19i)T + 121iT^{2} \)
13 \( 1 + (14.6 + 14.6i)T + 169iT^{2} \)
17 \( 1 - 28.0iT - 289T^{2} \)
19 \( 1 + (-12.5 - 12.5i)T + 361iT^{2} \)
23 \( 1 - 29.2T + 529T^{2} \)
29 \( 1 + (-19.3 + 19.3i)T - 841iT^{2} \)
31 \( 1 - 11.6T + 961T^{2} \)
37 \( 1 + (-0.771 + 0.771i)T - 1.36e3iT^{2} \)
41 \( 1 - 25.6T + 1.68e3T^{2} \)
43 \( 1 + (40.5 - 40.5i)T - 1.84e3iT^{2} \)
47 \( 1 - 50.2iT - 2.20e3T^{2} \)
53 \( 1 + (46.2 + 46.2i)T + 2.80e3iT^{2} \)
59 \( 1 + (22.7 + 22.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (-12.7 - 12.7i)T + 3.72e3iT^{2} \)
67 \( 1 + (-10.6 - 10.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 122.T + 5.04e3T^{2} \)
73 \( 1 - 15.0iT - 5.32e3T^{2} \)
79 \( 1 + 51.3T + 6.24e3T^{2} \)
83 \( 1 + (-37.8 + 37.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 5.45T + 7.92e3T^{2} \)
97 \( 1 + 81.1T + 9.40e3T^{2} \)
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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

−15.15427504548365684692349676988, −14.50823856765438232646977226401, −12.97134373960412429814567946575, −12.00198784808589904868854024475, −10.67258938371645493834598931413, −9.749021711169674979786544833317, −8.171561573859546981471822898196, −5.79456218876992632961998236109, −4.58093625976878605689822281084, −2.98152972252694508629880344077, 2.89884597339543381502222992146, 4.95915352064354997919134315054, 7.04628767577453036506155257882, 7.25868542493107346723300485411, 9.082009937432160813873889296034, 11.36101225933708535504914936878, 12.14701439728204063484676876124, 13.47360794784795308319376958565, 14.13357724261143228592232773272, 15.18145232914011619993552984083

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