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Properties

Label 2-48-12.11-c5-0-1
Degree $2$
Conductor $48$
Sign $0.896 - 0.442i$
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  + (−12.9 − 8.66i)3-s + 44.8i·5-s − 38.1i·7-s + (92.9 + 224. i)9-s + 544.·11-s + 814·13-s + (388. − 581. i)15-s + 1.97e3i·17-s + 38.1i·19-s + (−330 + 493. i)21-s − 1.39e3·23-s + 1.10e3·25-s + (738. − 3.71e3i)27-s − 5.43e3i·29-s + 1.04e4i·31-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)3-s + 0.803i·5-s − 0.293i·7-s + (0.382 + 0.923i)9-s + 1.35·11-s + 1.33·13-s + (0.446 − 0.667i)15-s + 1.65i·17-s + 0.0242i·19-s + (−0.163 + 0.244i)21-s − 0.551·23-s + 0.354·25-s + (0.195 − 0.980i)27-s − 1.19i·29-s + 1.94i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.896 - 0.442i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ 0.896 - 0.442i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.29310 + 0.301522i\)
\(L(\frac12)\) \(\approx\) \(1.29310 + 0.301522i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (12.9 + 8.66i)T \)
good5 \( 1 - 44.8iT - 3.12e3T^{2} \)
7 \( 1 + 38.1iT - 1.68e4T^{2} \)
11 \( 1 - 544.T + 1.61e5T^{2} \)
13 \( 1 - 814T + 3.71e5T^{2} \)
17 \( 1 - 1.97e3iT - 1.41e6T^{2} \)
19 \( 1 - 38.1iT - 2.47e6T^{2} \)
23 \( 1 + 1.39e3T + 6.43e6T^{2} \)
29 \( 1 + 5.43e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.04e4iT - 2.86e7T^{2} \)
37 \( 1 + 1.59e3T + 6.93e7T^{2} \)
41 \( 1 + 8.89e3iT - 1.15e8T^{2} \)
43 \( 1 - 9.03e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.64e4T + 2.29e8T^{2} \)
53 \( 1 - 1.92e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.56e4T + 7.14e8T^{2} \)
61 \( 1 - 2.38e4T + 8.44e8T^{2} \)
67 \( 1 + 3.03e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.28e4T + 1.80e9T^{2} \)
73 \( 1 - 1.75e4T + 2.07e9T^{2} \)
79 \( 1 - 7.12e3iT - 3.07e9T^{2} \)
83 \( 1 - 1.62e4T + 3.93e9T^{2} \)
89 \( 1 + 8.84e4iT - 5.58e9T^{2} \)
97 \( 1 + 4.90e4T + 8.58e9T^{2} \)
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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

−14.60992678679024764112242649150, −13.56513464143422115168479134488, −12.30219323597743753176941769559, −11.17754723842563833161531769469, −10.35186458525617025511995313036, −8.434703029935748316592360258113, −6.82509346594668318358843787237, −6.05960487596838311017661844456, −3.89586078918740177900650871858, −1.41482110293227740910872288535, 0.935959272208809740344281422815, 3.94754323134692544541969154430, 5.33811882272077207077625118109, 6.65584359787269791099288835163, 8.754173856698912193403029870706, 9.641572493361603677967196774603, 11.28015789517888498567951961041, 11.96701763083631249453347395868, 13.29126690007968615111244993841, 14.71443524413297407285122068222

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