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Properties

Label 2-48-12.11-c21-0-22
Degree $2$
Conductor $48$
Sign $0.497 - 0.867i$
Analytic cond. $134.149$
Root an. cond. $11.5822$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.13e4 + 8.84e4i)3-s − 1.40e6i·5-s + 5.11e8i·7-s + (−5.18e9 + 9.08e9i)9-s − 1.07e11·11-s + 8.24e11·13-s + (1.24e11 − 7.21e10i)15-s − 1.34e13i·17-s − 2.46e13i·19-s + (−4.52e13 + 2.62e13i)21-s + 3.24e14·23-s + 4.74e14·25-s + (−1.06e15 + 8.40e12i)27-s + 3.04e15i·29-s − 3.02e15i·31-s + ⋯
L(s)  = 1  + (0.502 + 0.864i)3-s − 0.0643i·5-s + 0.683i·7-s + (−0.495 + 0.868i)9-s − 1.25·11-s + 1.65·13-s + (0.0556 − 0.0323i)15-s − 1.62i·17-s − 0.923i·19-s + (−0.591 + 0.343i)21-s + 1.63·23-s + 0.995·25-s + (−0.999 + 0.00785i)27-s + 1.34i·29-s − 0.662i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.497 - 0.867i$
Analytic conductor: \(134.149\)
Root analytic conductor: \(11.5822\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :21/2),\ 0.497 - 0.867i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.820464997\)
\(L(\frac12)\) \(\approx\) \(2.820464997\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-5.13e4 - 8.84e4i)T \)
good5 \( 1 + 1.40e6iT - 4.76e14T^{2} \)
7 \( 1 - 5.11e8iT - 5.58e17T^{2} \)
11 \( 1 + 1.07e11T + 7.40e21T^{2} \)
13 \( 1 - 8.24e11T + 2.47e23T^{2} \)
17 \( 1 + 1.34e13iT - 6.90e25T^{2} \)
19 \( 1 + 2.46e13iT - 7.14e26T^{2} \)
23 \( 1 - 3.24e14T + 3.94e28T^{2} \)
29 \( 1 - 3.04e15iT - 5.13e30T^{2} \)
31 \( 1 + 3.02e15iT - 2.08e31T^{2} \)
37 \( 1 - 1.03e16T + 8.55e32T^{2} \)
41 \( 1 - 6.41e16iT - 7.38e33T^{2} \)
43 \( 1 + 2.36e17iT - 2.00e34T^{2} \)
47 \( 1 - 3.36e16T + 1.30e35T^{2} \)
53 \( 1 - 1.87e18iT - 1.62e36T^{2} \)
59 \( 1 - 6.89e18T + 1.54e37T^{2} \)
61 \( 1 + 3.93e18T + 3.10e37T^{2} \)
67 \( 1 + 9.06e18iT - 2.22e38T^{2} \)
71 \( 1 - 2.00e18T + 7.52e38T^{2} \)
73 \( 1 + 7.17e18T + 1.34e39T^{2} \)
79 \( 1 + 4.19e19iT - 7.08e39T^{2} \)
83 \( 1 + 1.64e20T + 1.99e40T^{2} \)
89 \( 1 - 1.53e20iT - 8.65e40T^{2} \)
97 \( 1 + 4.21e20T + 5.27e41T^{2} \)
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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

−11.31687744856681338377771739386, −10.62044583948401362670658711827, −9.146665116818101032640356025986, −8.642887427268918473111195575250, −7.19788367146515092057660350628, −5.50122883376775591243320611780, −4.76577173756615250597736873749, −3.19401351547254917020645963573, −2.54909932045476781499112581419, −0.799448352565570028834746349548, 0.74093176702862247550326641214, 1.58669919411404173006511917162, 2.95239868091503438325937499568, 3.95851482889487613457723248438, 5.72572555317474887731291682350, 6.75322215514397959263315579082, 7.969653556874521352135271733480, 8.629857999484848927964905440512, 10.30596050268981455347572955857, 11.20093054103178535967403348771

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