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Properties

Label 2-48-3.2-c8-0-1
Degree $2$
Conductor $48$
Sign $-0.996 - 0.0805i$
Analytic cond. $19.5541$
Root an. cond. $4.42201$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−80.7 − 6.52i)3-s + 920. i·5-s + 2.01e3·7-s + (6.47e3 + 1.05e3i)9-s − 9.96e3i·11-s − 1.12e4·13-s + (6.00e3 − 7.43e4i)15-s + 1.51e5i·17-s − 1.86e5·19-s + (−1.62e5 − 1.31e4i)21-s + 2.24e5i·23-s − 4.56e5·25-s + (−5.15e5 − 1.27e5i)27-s − 1.17e6i·29-s − 3.00e5·31-s + ⋯
L(s)  = 1  + (−0.996 − 0.0805i)3-s + 1.47i·5-s + 0.838·7-s + (0.987 + 0.160i)9-s − 0.680i·11-s − 0.395·13-s + (0.118 − 1.46i)15-s + 1.81i·17-s − 1.42·19-s + (−0.835 − 0.0675i)21-s + 0.803i·23-s − 1.16·25-s + (−0.970 − 0.239i)27-s − 1.66i·29-s − 0.325·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0805i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.996 - 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.996 - 0.0805i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ -0.996 - 0.0805i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0218181 + 0.540627i\)
\(L(\frac12)\) \(\approx\) \(0.0218181 + 0.540627i\)
\(L(5)\) 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 \( 1 + (80.7 + 6.52i)T \)
good5 \( 1 - 920. iT - 3.90e5T^{2} \)
7 \( 1 - 2.01e3T + 5.76e6T^{2} \)
11 \( 1 + 9.96e3iT - 2.14e8T^{2} \)
13 \( 1 + 1.12e4T + 8.15e8T^{2} \)
17 \( 1 - 1.51e5iT - 6.97e9T^{2} \)
19 \( 1 + 1.86e5T + 1.69e10T^{2} \)
23 \( 1 - 2.24e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.17e6iT - 5.00e11T^{2} \)
31 \( 1 + 3.00e5T + 8.52e11T^{2} \)
37 \( 1 + 1.47e6T + 3.51e12T^{2} \)
41 \( 1 + 2.69e6iT - 7.98e12T^{2} \)
43 \( 1 + 5.09e6T + 1.16e13T^{2} \)
47 \( 1 + 2.64e6iT - 2.38e13T^{2} \)
53 \( 1 - 3.07e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.78e7iT - 1.46e14T^{2} \)
61 \( 1 - 8.90e6T + 1.91e14T^{2} \)
67 \( 1 - 1.07e7T + 4.06e14T^{2} \)
71 \( 1 + 1.05e6iT - 6.45e14T^{2} \)
73 \( 1 - 1.03e7T + 8.06e14T^{2} \)
79 \( 1 + 6.99e7T + 1.51e15T^{2} \)
83 \( 1 + 2.34e7iT - 2.25e15T^{2} \)
89 \( 1 - 7.62e7iT - 3.93e15T^{2} \)
97 \( 1 - 4.70e7T + 7.83e15T^{2} \)
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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

−14.69396708027497006971213899586, −13.31820238901806276022246353904, −11.85253053712985340447470818330, −10.90310007523395691299998972885, −10.29130642027342955584420769331, −8.126517881672147968996962940842, −6.77463477487743621562420961955, −5.75992564579962198409144864923, −3.95729006396511968223032149500, −1.92437111733589531543048990536, 0.22270034388791741990401513622, 1.61140453448965082026344766633, 4.66530133397094006089770625338, 5.07248847129410807626901680608, 6.95075863627761074249342944309, 8.494709664291421441778163552569, 9.759756214103642366532745774640, 11.18793486905220180254280736097, 12.24765804439424539512088021996, 12.95044538765936803305552632429

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