Properties

Label 2-48-16.3-c4-0-1
Degree $2$
Conductor $48$
Sign $-0.515 + 0.856i$
Analytic cond. $4.96175$
Root an. cond. $2.22750$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.149 + 3.99i)2-s + (−3.67 + 3.67i)3-s + (−15.9 − 1.19i)4-s + (−0.419 + 0.419i)5-s + (−14.1 − 15.2i)6-s − 40.4·7-s + (7.15 − 63.5i)8-s − 27i·9-s + (−1.61 − 1.74i)10-s + (−67.0 − 67.0i)11-s + (63.0 − 54.2i)12-s + (22.3 + 22.3i)13-s + (6.03 − 161. i)14-s − 3.08i·15-s + (253. + 38.1i)16-s − 171.·17-s + ⋯
L(s)  = 1  + (−0.0373 + 0.999i)2-s + (−0.408 + 0.408i)3-s + (−0.997 − 0.0746i)4-s + (−0.0167 + 0.0167i)5-s + (−0.392 − 0.423i)6-s − 0.824·7-s + (0.111 − 0.993i)8-s − 0.333i·9-s + (−0.0161 − 0.0174i)10-s + (−0.554 − 0.554i)11-s + (0.437 − 0.376i)12-s + (0.132 + 0.132i)13-s + (0.0308 − 0.824i)14-s − 0.0137i·15-s + (0.988 + 0.148i)16-s − 0.593·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.515 + 0.856i$
Analytic conductor: \(4.96175\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :2),\ -0.515 + 0.856i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.104228 - 0.184464i\)
\(L(\frac12)\) \(\approx\) \(0.104228 - 0.184464i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.149 - 3.99i)T \)
3 \( 1 + (3.67 - 3.67i)T \)
good5 \( 1 + (0.419 - 0.419i)T - 625iT^{2} \)
7 \( 1 + 40.4T + 2.40e3T^{2} \)
11 \( 1 + (67.0 + 67.0i)T + 1.46e4iT^{2} \)
13 \( 1 + (-22.3 - 22.3i)T + 2.85e4iT^{2} \)
17 \( 1 + 171.T + 8.35e4T^{2} \)
19 \( 1 + (309. - 309. i)T - 1.30e5iT^{2} \)
23 \( 1 + 763.T + 2.79e5T^{2} \)
29 \( 1 + (-16.3 - 16.3i)T + 7.07e5iT^{2} \)
31 \( 1 - 1.32e3iT - 9.23e5T^{2} \)
37 \( 1 + (-1.41e3 + 1.41e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 388. iT - 2.82e6T^{2} \)
43 \( 1 + (611. + 611. i)T + 3.41e6iT^{2} \)
47 \( 1 - 317. iT - 4.87e6T^{2} \)
53 \( 1 + (-2.46e3 + 2.46e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (3.64e3 + 3.64e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-2.83e3 - 2.83e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (3.39e3 - 3.39e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 6.50e3T + 2.54e7T^{2} \)
73 \( 1 - 710. iT - 2.83e7T^{2} \)
79 \( 1 - 9.85e3iT - 3.89e7T^{2} \)
83 \( 1 + (-8.86e3 + 8.86e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.16e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.77e4T + 8.85e7T^{2} \)
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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

−15.95550350091122592329605696625, −14.76012990959810387231307904880, −13.52583777095001582594162675305, −12.46330926869517481635604081541, −10.68834121655161706093990454231, −9.551103163197223017506785725588, −8.262001420037914723362675666743, −6.64973178383585586350874276549, −5.56344319339166490490035709920, −3.86048327734278042121576422702, 0.13404875810315423478941756141, 2.41705466961108897536318516142, 4.42902276428576722567291256871, 6.24167785895161683888218642970, 8.058701138368987875850581228835, 9.592258084754448252043929553769, 10.63133975171488823725057663736, 11.86557865575449711500130998979, 12.87896476878748488973624992988, 13.59344169393240840173210378077

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