Properties

Label 2-48-3.2-c8-0-5
Degree $2$
Conductor $48$
Sign $0.555 - 0.831i$
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  + (−45 + 67.3i)3-s − 224. i·5-s + 1.75e3·7-s + (−2.51e3 − 6.06e3i)9-s − 6.95e3i·11-s + 2.57e4·13-s + (1.51e4 + 1.01e4i)15-s + 7.48e4i·17-s − 1.89e4·19-s + (−7.87e4 + 1.17e5i)21-s + 4.70e5i·23-s + 3.40e5·25-s + (5.21e5 + 1.03e5i)27-s + 4.60e5i·29-s + 3.51e5·31-s + ⋯
L(s)  = 1  + (−0.555 + 0.831i)3-s − 0.359i·5-s + 0.728·7-s + (−0.382 − 0.923i)9-s − 0.475i·11-s + 0.900·13-s + (0.298 + 0.199i)15-s + 0.896i·17-s − 0.145·19-s + (−0.404 + 0.606i)21-s + 1.68i·23-s + 0.870·25-s + (0.980 + 0.195i)27-s + 0.651i·29-s + 0.380·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.555 - 0.831i$
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: \((2,\ 48,\ (\ :4),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.45789 + 0.779276i\)
\(L(\frac12)\) \(\approx\) \(1.45789 + 0.779276i\)
\(L(5)\) 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 \( 1 + (45 - 67.3i)T \)
good5 \( 1 + 224. iT - 3.90e5T^{2} \)
7 \( 1 - 1.75e3T + 5.76e6T^{2} \)
11 \( 1 + 6.95e3iT - 2.14e8T^{2} \)
13 \( 1 - 2.57e4T + 8.15e8T^{2} \)
17 \( 1 - 7.48e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.89e4T + 1.69e10T^{2} \)
23 \( 1 - 4.70e5iT - 7.83e10T^{2} \)
29 \( 1 - 4.60e5iT - 5.00e11T^{2} \)
31 \( 1 - 3.51e5T + 8.52e11T^{2} \)
37 \( 1 - 1.33e6T + 3.51e12T^{2} \)
41 \( 1 - 1.87e6iT - 7.98e12T^{2} \)
43 \( 1 - 3.52e6T + 1.16e13T^{2} \)
47 \( 1 - 4.08e6iT - 2.38e13T^{2} \)
53 \( 1 + 6.60e6iT - 6.22e13T^{2} \)
59 \( 1 + 1.37e7iT - 1.46e14T^{2} \)
61 \( 1 - 7.53e5T + 1.91e14T^{2} \)
67 \( 1 + 2.26e6T + 4.06e14T^{2} \)
71 \( 1 + 1.70e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.76e7T + 8.06e14T^{2} \)
79 \( 1 - 2.29e7T + 1.51e15T^{2} \)
83 \( 1 - 4.63e7iT - 2.25e15T^{2} \)
89 \( 1 - 7.26e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.47e8T + 7.83e15T^{2} \)
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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

−14.27448520457435243076198718689, −12.86274930643841808110116366317, −11.46640651235785166298089456403, −10.76121746546170846429735256890, −9.310256699504618723072463268847, −8.172804267250429017577728294865, −6.17716237888234916682840836213, −4.98065529848868112208831702534, −3.61288252650347321324123224000, −1.15754116742170621321686701466, 0.817325643783789336294085270175, 2.40001692032778338434242710382, 4.68812500496455518655213585337, 6.20578692927325601483921042856, 7.36919042493287258435336054758, 8.612045762433163807982767093047, 10.51076159371345303411999712944, 11.43264430915779336999848573295, 12.51277768015357918966946082947, 13.70260107437466586894635630160

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