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Properties

Label 2-48-3.2-c4-0-5
Degree $2$
Conductor $48$
Sign $-0.690 + 0.723i$
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  + (−6.21 + 6.51i)3-s − 16.4i·5-s − 49.2·7-s + (−3.84 − 80.9i)9-s − 207. i·11-s − 235.·13-s + (107. + 102. i)15-s + 337. i·17-s − 162.·19-s + (306 − 320. i)21-s + 185. i·23-s + 354.·25-s + (550. + 477. i)27-s − 361. i·29-s − 474.·31-s + ⋯
L(s)  = 1  + (−0.690 + 0.723i)3-s − 0.658i·5-s − 1.00·7-s + (−0.0474 − 0.998i)9-s − 1.71i·11-s − 1.39·13-s + (0.476 + 0.454i)15-s + 1.16i·17-s − 0.450·19-s + (0.693 − 0.727i)21-s + 0.349i·23-s + 0.566·25-s + (0.755 + 0.654i)27-s − 0.430i·29-s − 0.493·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.690 + 0.723i$
Analytic conductor: \(4.96175\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :2),\ -0.690 + 0.723i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.137124 - 0.320243i\)
\(L(\frac12)\) \(\approx\) \(0.137124 - 0.320243i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (6.21 - 6.51i)T \)
good5 \( 1 + 16.4iT - 625T^{2} \)
7 \( 1 + 49.2T + 2.40e3T^{2} \)
11 \( 1 + 207. iT - 1.46e4T^{2} \)
13 \( 1 + 235.T + 2.85e4T^{2} \)
17 \( 1 - 337. iT - 8.35e4T^{2} \)
19 \( 1 + 162.T + 1.30e5T^{2} \)
23 \( 1 - 185. iT - 2.79e5T^{2} \)
29 \( 1 + 361. iT - 7.07e5T^{2} \)
31 \( 1 + 474.T + 9.23e5T^{2} \)
37 \( 1 + 1.91e3T + 1.87e6T^{2} \)
41 \( 1 - 460. iT - 2.82e6T^{2} \)
43 \( 1 + 60.8T + 3.41e6T^{2} \)
47 \( 1 + 238. iT - 4.87e6T^{2} \)
53 \( 1 + 5.28e3iT - 7.89e6T^{2} \)
59 \( 1 + 306. iT - 1.21e7T^{2} \)
61 \( 1 - 242.T + 1.38e7T^{2} \)
67 \( 1 - 3.99e3T + 2.01e7T^{2} \)
71 \( 1 + 908. iT - 2.54e7T^{2} \)
73 \( 1 - 3.45e3T + 2.83e7T^{2} \)
79 \( 1 - 6.28e3T + 3.89e7T^{2} \)
83 \( 1 + 5.13e3iT - 4.74e7T^{2} \)
89 \( 1 + 4.66e3iT - 6.27e7T^{2} \)
97 \( 1 - 3.78e3T + 8.85e7T^{2} \)
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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

−14.66887576988759142356817202567, −13.10664400933301373689669067665, −12.16103068132010529266391818836, −10.84330390249714442441000052506, −9.725723068801479128936419073226, −8.570718314209767436795824291181, −6.46048965694804543777575201918, −5.25539272040424110299287063076, −3.55504781537720858176141800861, −0.22054334635729659442557635091, 2.46238957844519133733820762097, 4.95980456104048854719918554999, 6.78456308949997375857755287549, 7.29443601192892709898381682623, 9.568408310604394007115582192537, 10.59027246573750934093601213334, 12.16032662274416472457276720059, 12.68564896612028495898491288340, 14.14306688365056009487321062255, 15.31749697448597674317187485296

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