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Properties

Label 2-48-16.5-c5-0-7
Degree $2$
Conductor $48$
Sign $0.230 - 0.973i$
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  + (5.58 + 0.880i)2-s + (−6.36 + 6.36i)3-s + (30.4 + 9.84i)4-s + (3.64 + 3.64i)5-s + (−41.1 + 29.9i)6-s + 162. i·7-s + (161. + 81.8i)8-s − 81i·9-s + (17.1 + 23.5i)10-s + (195. + 195. i)11-s + (−256. + 131. i)12-s + (−264. + 264. i)13-s + (−143. + 909. i)14-s − 46.4·15-s + (830. + 599. i)16-s − 55.8·17-s + ⋯
L(s)  = 1  + (0.987 + 0.155i)2-s + (−0.408 + 0.408i)3-s + (0.951 + 0.307i)4-s + (0.0652 + 0.0652i)5-s + (−0.466 + 0.339i)6-s + 1.25i·7-s + (0.892 + 0.451i)8-s − 0.333i·9-s + (0.0542 + 0.0745i)10-s + (0.488 + 0.488i)11-s + (−0.514 + 0.262i)12-s + (−0.434 + 0.434i)13-s + (−0.195 + 1.23i)14-s − 0.0532·15-s + (0.810 + 0.585i)16-s − 0.0469·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.230 - 0.973i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ 0.230 - 0.973i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.08079 + 1.64543i\)
\(L(\frac12)\) \(\approx\) \(2.08079 + 1.64543i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.58 - 0.880i)T \)
3 \( 1 + (6.36 - 6.36i)T \)
good5 \( 1 + (-3.64 - 3.64i)T + 3.12e3iT^{2} \)
7 \( 1 - 162. iT - 1.68e4T^{2} \)
11 \( 1 + (-195. - 195. i)T + 1.61e5iT^{2} \)
13 \( 1 + (264. - 264. i)T - 3.71e5iT^{2} \)
17 \( 1 + 55.8T + 1.41e6T^{2} \)
19 \( 1 + (-693. + 693. i)T - 2.47e6iT^{2} \)
23 \( 1 + 2.42e3iT - 6.43e6T^{2} \)
29 \( 1 + (-958. + 958. i)T - 2.05e7iT^{2} \)
31 \( 1 - 5.30e3T + 2.86e7T^{2} \)
37 \( 1 + (9.95e3 + 9.95e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 2.15e3iT - 1.15e8T^{2} \)
43 \( 1 + (1.29e4 + 1.29e4i)T + 1.47e8iT^{2} \)
47 \( 1 - 2.50e4T + 2.29e8T^{2} \)
53 \( 1 + (-5.46e3 - 5.46e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (-2.71e4 - 2.71e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (4.01e4 - 4.01e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (559. - 559. i)T - 1.35e9iT^{2} \)
71 \( 1 - 5.22e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.75e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.10e4T + 3.07e9T^{2} \)
83 \( 1 + (1.80e4 - 1.80e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 9.72e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.00e5T + 8.58e9T^{2} \)
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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

−14.97832622109537805693801907957, −13.90170946825511080512024525831, −12.26827439094526928365653458784, −11.88844713673583400938576300643, −10.36592559273806927739073757794, −8.807326288768661173804717859485, −6.89701925010158545266639659758, −5.66385968222575510493723379409, −4.40349419543405251731650691392, −2.46169876621335670595902386885, 1.21021491451988075748041648369, 3.51345923237393927570819225433, 5.11891380471264055521183020691, 6.58394243686926351159672381256, 7.68870708047634746991422977095, 10.03005951495655465892417297045, 11.13343290396761479570600349535, 12.16465252904423367720027678308, 13.42144100900459949085890865464, 14.00691431175419050944050384267

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