Properties

Label 2-48-1.1-c27-0-10
Degree $2$
Conductor $48$
Sign $1$
Analytic cond. $221.690$
Root an. cond. $14.8892$
Motivic weight $27$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.59e6·3-s − 2.94e9·5-s + 2.52e11·7-s + 2.54e12·9-s + 1.51e14·11-s + 9.28e13·13-s + 4.70e15·15-s + 1.39e16·17-s + 1.70e17·19-s − 4.03e17·21-s + 4.09e18·23-s + 1.24e18·25-s − 4.05e18·27-s + 4.32e19·29-s + 1.99e20·31-s − 2.41e20·33-s − 7.45e20·35-s + 1.37e21·37-s − 1.47e20·39-s + 6.82e21·41-s − 9.39e21·43-s − 7.49e21·45-s + 2.13e21·47-s − 1.77e21·49-s − 2.21e22·51-s − 2.93e23·53-s − 4.46e23·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.08·5-s + 0.986·7-s + 0.333·9-s + 1.32·11-s + 0.0849·13-s + 0.623·15-s + 0.340·17-s + 0.931·19-s − 0.569·21-s + 1.69·23-s + 0.167·25-s − 0.192·27-s + 0.782·29-s + 1.46·31-s − 0.763·33-s − 1.06·35-s + 0.929·37-s − 0.0490·39-s + 1.15·41-s − 0.833·43-s − 0.360·45-s + 0.0570·47-s − 0.0270·49-s − 0.196·51-s − 1.54·53-s − 1.42·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $1$
Analytic conductor: \(221.690\)
Root analytic conductor: \(14.8892\)
Motivic weight: \(27\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(2.352759572\)
\(L(\frac12)\) \(\approx\) \(2.352759572\)
\(L(\frac{29}{2})\) 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 \( 1 + 1.59e6T \)
good5 \( 1 + 2.94e9T + 7.45e18T^{2} \)
7 \( 1 - 2.52e11T + 6.57e22T^{2} \)
11 \( 1 - 1.51e14T + 1.31e28T^{2} \)
13 \( 1 - 9.28e13T + 1.19e30T^{2} \)
17 \( 1 - 1.39e16T + 1.66e33T^{2} \)
19 \( 1 - 1.70e17T + 3.36e34T^{2} \)
23 \( 1 - 4.09e18T + 5.84e36T^{2} \)
29 \( 1 - 4.32e19T + 3.05e39T^{2} \)
31 \( 1 - 1.99e20T + 1.84e40T^{2} \)
37 \( 1 - 1.37e21T + 2.19e42T^{2} \)
41 \( 1 - 6.82e21T + 3.50e43T^{2} \)
43 \( 1 + 9.39e21T + 1.26e44T^{2} \)
47 \( 1 - 2.13e21T + 1.40e45T^{2} \)
53 \( 1 + 2.93e23T + 3.59e46T^{2} \)
59 \( 1 + 8.14e23T + 6.50e47T^{2} \)
61 \( 1 - 5.59e23T + 1.59e48T^{2} \)
67 \( 1 - 6.51e24T + 2.01e49T^{2} \)
71 \( 1 + 3.69e24T + 9.63e49T^{2} \)
73 \( 1 - 1.27e24T + 2.04e50T^{2} \)
79 \( 1 - 1.00e25T + 1.72e51T^{2} \)
83 \( 1 + 9.33e25T + 6.53e51T^{2} \)
89 \( 1 - 3.02e25T + 4.30e52T^{2} \)
97 \( 1 + 6.11e26T + 4.39e53T^{2} \)
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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

−11.16661080516469986872973499085, −9.624330963425660591378496626782, −8.372824041828297423657529897619, −7.44754266522359880973419141283, −6.39005105666963346742910720730, −4.97957190312824053769100598158, −4.23935200993842494788153466615, −3.07960985486359999548074904911, −1.33741138524817812724328805460, −0.76587562504596423637576007514, 0.76587562504596423637576007514, 1.33741138524817812724328805460, 3.07960985486359999548074904911, 4.23935200993842494788153466615, 4.97957190312824053769100598158, 6.39005105666963346742910720730, 7.44754266522359880973419141283, 8.372824041828297423657529897619, 9.624330963425660591378496626782, 11.16661080516469986872973499085

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