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Properties

Label 2-48-3.2-c10-0-2
Degree $2$
Conductor $48$
Sign $-0.996 - 0.0775i$
Analytic cond. $30.4971$
Root an. cond. $5.52242$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−242. − 18.8i)3-s + 4.81e3i·5-s − 670.·7-s + (5.83e4 + 9.12e3i)9-s + 2.33e5i·11-s + 3.07e5·13-s + (9.07e4 − 1.16e6i)15-s − 6.72e5i·17-s + 1.55e6·19-s + (1.62e5 + 1.26e4i)21-s + 5.57e6i·23-s − 1.34e7·25-s + (−1.39e7 − 3.30e6i)27-s + 2.97e7i·29-s − 3.09e7·31-s + ⋯
L(s)  = 1  + (−0.996 − 0.0775i)3-s + 1.54i·5-s − 0.0398·7-s + (0.987 + 0.154i)9-s + 1.44i·11-s + 0.828·13-s + (0.119 − 1.53i)15-s − 0.473i·17-s + 0.626·19-s + (0.0397 + 0.00309i)21-s + 0.866i·23-s − 1.37·25-s + (−0.973 − 0.230i)27-s + 1.44i·29-s − 1.08·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0775i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.996 - 0.0775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.996 - 0.0775i$
Analytic conductor: \(30.4971\)
Root analytic conductor: \(5.52242\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5),\ -0.996 - 0.0775i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0332372 + 0.856401i\)
\(L(\frac12)\) \(\approx\) \(0.0332372 + 0.856401i\)
\(L(6)\) 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 \( 1 + (242. + 18.8i)T \)
good5 \( 1 - 4.81e3iT - 9.76e6T^{2} \)
7 \( 1 + 670.T + 2.82e8T^{2} \)
11 \( 1 - 2.33e5iT - 2.59e10T^{2} \)
13 \( 1 - 3.07e5T + 1.37e11T^{2} \)
17 \( 1 + 6.72e5iT - 2.01e12T^{2} \)
19 \( 1 - 1.55e6T + 6.13e12T^{2} \)
23 \( 1 - 5.57e6iT - 4.14e13T^{2} \)
29 \( 1 - 2.97e7iT - 4.20e14T^{2} \)
31 \( 1 + 3.09e7T + 8.19e14T^{2} \)
37 \( 1 + 8.56e7T + 4.80e15T^{2} \)
41 \( 1 + 3.59e7iT - 1.34e16T^{2} \)
43 \( 1 - 3.66e7T + 2.16e16T^{2} \)
47 \( 1 - 3.28e7iT - 5.25e16T^{2} \)
53 \( 1 + 4.59e8iT - 1.74e17T^{2} \)
59 \( 1 + 4.88e8iT - 5.11e17T^{2} \)
61 \( 1 + 6.12e7T + 7.13e17T^{2} \)
67 \( 1 - 6.70e8T + 1.82e18T^{2} \)
71 \( 1 - 1.23e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.08e9T + 4.29e18T^{2} \)
79 \( 1 - 1.86e9T + 9.46e18T^{2} \)
83 \( 1 + 1.09e9iT - 1.55e19T^{2} \)
89 \( 1 + 5.19e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.07e10T + 7.37e19T^{2} \)
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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

−14.11110583280982446274247841838, −12.71523732368757002170418431299, −11.50329862126986267706065344063, −10.67291073891345609025294735275, −9.662290241644942840000440977432, −7.35088189872933118606006833507, −6.72149989596327304784046718030, −5.26350446563490129896014607466, −3.52602189511198974885081984372, −1.73057288880771767552257724014, 0.33800691113948967619830486248, 1.27582901930568280889361648610, 3.92310508450966843554732229668, 5.26422637021701204908097444325, 6.18125350502726194287450708420, 8.144580555360930798967396342517, 9.207839768619646266192735415022, 10.72594344060500469217457782590, 11.78517088892009284509895662164, 12.80616014171487785248682279462

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