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Properties

Label 2-48-16.13-c3-0-0
Degree $2$
Conductor $48$
Sign $-0.373 - 0.927i$
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  + (−2.77 + 0.544i)2-s + (−2.12 − 2.12i)3-s + (7.40 − 3.02i)4-s + (−3.72 + 3.72i)5-s + (7.04 + 4.73i)6-s + 20.2i·7-s + (−18.9 + 12.4i)8-s + 8.99i·9-s + (8.30 − 12.3i)10-s + (−47.5 + 47.5i)11-s + (−22.1 − 9.30i)12-s + (27.8 + 27.8i)13-s + (−11.0 − 56.2i)14-s + 15.8·15-s + (45.7 − 44.7i)16-s + 56.3·17-s + ⋯
L(s)  = 1  + (−0.981 + 0.192i)2-s + (−0.408 − 0.408i)3-s + (0.925 − 0.377i)4-s + (−0.333 + 0.333i)5-s + (0.479 + 0.322i)6-s + 1.09i·7-s + (−0.835 + 0.549i)8-s + 0.333i·9-s + (0.262 − 0.390i)10-s + (−1.30 + 1.30i)11-s + (−0.532 − 0.223i)12-s + (0.594 + 0.594i)13-s + (−0.210 − 1.07i)14-s + 0.271·15-s + (0.714 − 0.699i)16-s + 0.804·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.373 - 0.927i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.373 - 0.927i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.280132 + 0.414542i\)
\(L(\frac12)\) \(\approx\) \(0.280132 + 0.414542i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.77 - 0.544i)T \)
3 \( 1 + (2.12 + 2.12i)T \)
good5 \( 1 + (3.72 - 3.72i)T - 125iT^{2} \)
7 \( 1 - 20.2iT - 343T^{2} \)
11 \( 1 + (47.5 - 47.5i)T - 1.33e3iT^{2} \)
13 \( 1 + (-27.8 - 27.8i)T + 2.19e3iT^{2} \)
17 \( 1 - 56.3T + 4.91e3T^{2} \)
19 \( 1 + (66.1 + 66.1i)T + 6.85e3iT^{2} \)
23 \( 1 + 3.66iT - 1.21e4T^{2} \)
29 \( 1 + (86.8 + 86.8i)T + 2.43e4iT^{2} \)
31 \( 1 + 102.T + 2.97e4T^{2} \)
37 \( 1 + (-66.8 + 66.8i)T - 5.06e4iT^{2} \)
41 \( 1 - 29.5iT - 6.89e4T^{2} \)
43 \( 1 + (372. - 372. i)T - 7.95e4iT^{2} \)
47 \( 1 - 539.T + 1.03e5T^{2} \)
53 \( 1 + (-385. + 385. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-71.0 + 71.0i)T - 2.05e5iT^{2} \)
61 \( 1 + (155. + 155. i)T + 2.26e5iT^{2} \)
67 \( 1 + (178. + 178. i)T + 3.00e5iT^{2} \)
71 \( 1 - 483. iT - 3.57e5T^{2} \)
73 \( 1 - 908. iT - 3.89e5T^{2} \)
79 \( 1 - 1.06e3T + 4.93e5T^{2} \)
83 \( 1 + (-871. - 871. i)T + 5.71e5iT^{2} \)
89 \( 1 - 185. iT - 7.04e5T^{2} \)
97 \( 1 + 725.T + 9.12e5T^{2} \)
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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

−15.53756232442483374930071424056, −14.93966203961712455220768144790, −12.91971100845159464964554441771, −11.81517474326529545304411887603, −10.78341954044095972615384453112, −9.446018220603328560716494940636, −8.054654363188834124389957759778, −6.93308958217034615709423478123, −5.46975436606283829916662192638, −2.26532416575940173535285636014, 0.53616721547059180945511268761, 3.56955489347725947388781698074, 5.81468265769789500555815427345, 7.62112510334979441770307096198, 8.605983792444969481417610861585, 10.45362353121419650593862738293, 10.68673022304207456231464449795, 12.20465751813453828130504666237, 13.51181216432814039069245591776, 15.26838941225671780699315365880

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