Properties

Label 2-48-1.1-c9-0-7
Degree $2$
Conductor $48$
Sign $-1$
Analytic cond. $24.7217$
Root an. cond. $4.97209$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81·3-s − 1.31e3·5-s + 4.48e3·7-s + 6.56e3·9-s − 1.47e3·11-s − 1.51e5·13-s − 1.06e5·15-s + 1.08e5·17-s − 5.93e5·19-s + 3.62e5·21-s + 9.69e5·23-s − 2.26e5·25-s + 5.31e5·27-s − 6.64e6·29-s − 7.07e6·31-s − 1.19e5·33-s − 5.88e6·35-s − 7.47e6·37-s − 1.22e7·39-s − 4.35e6·41-s + 4.35e6·43-s − 8.62e6·45-s − 2.83e7·47-s − 2.02e7·49-s + 8.76e6·51-s + 1.61e7·53-s + 1.93e6·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.940·5-s + 0.705·7-s + 1/3·9-s − 0.0303·11-s − 1.47·13-s − 0.542·15-s + 0.314·17-s − 1.04·19-s + 0.407·21-s + 0.722·23-s − 0.115·25-s + 0.192·27-s − 1.74·29-s − 1.37·31-s − 0.0175·33-s − 0.663·35-s − 0.655·37-s − 0.849·39-s − 0.240·41-s + 0.194·43-s − 0.313·45-s − 0.846·47-s − 0.502·49-s + 0.181·51-s + 0.280·53-s + 0.0285·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-1$
Analytic conductor: \(24.7217\)
Root analytic conductor: \(4.97209\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 48,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - p^{4} T \)
good5 \( 1 + 1314 T + p^{9} T^{2} \)
7 \( 1 - 640 p T + p^{9} T^{2} \)
11 \( 1 + 1476 T + p^{9} T^{2} \)
13 \( 1 + 151522 T + p^{9} T^{2} \)
17 \( 1 - 108162 T + p^{9} T^{2} \)
19 \( 1 + 593084 T + p^{9} T^{2} \)
23 \( 1 - 969480 T + p^{9} T^{2} \)
29 \( 1 + 6642522 T + p^{9} T^{2} \)
31 \( 1 + 7070600 T + p^{9} T^{2} \)
37 \( 1 + 7472410 T + p^{9} T^{2} \)
41 \( 1 + 4350150 T + p^{9} T^{2} \)
43 \( 1 - 4358716 T + p^{9} T^{2} \)
47 \( 1 + 28309248 T + p^{9} T^{2} \)
53 \( 1 - 16111710 T + p^{9} T^{2} \)
59 \( 1 - 86075964 T + p^{9} T^{2} \)
61 \( 1 - 32213918 T + p^{9} T^{2} \)
67 \( 1 + 99531452 T + p^{9} T^{2} \)
71 \( 1 - 44170488 T + p^{9} T^{2} \)
73 \( 1 + 23560630 T + p^{9} T^{2} \)
79 \( 1 - 401754760 T + p^{9} T^{2} \)
83 \( 1 - 744528708 T + p^{9} T^{2} \)
89 \( 1 - 769871034 T + p^{9} T^{2} \)
97 \( 1 - 907130882 T + p^{9} T^{2} \)
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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

−13.06267565558064360819841292317, −11.97216163855752854798208673036, −10.82845430029998821706160804127, −9.335605634997783277221241487177, −8.042785218603163251396388606649, −7.20541553487380733467907169907, −5.03647276094300303888600772149, −3.70994467330358667034517268199, −2.03244412601164472223955114225, 0, 2.03244412601164472223955114225, 3.70994467330358667034517268199, 5.03647276094300303888600772149, 7.20541553487380733467907169907, 8.042785218603163251396388606649, 9.335605634997783277221241487177, 10.82845430029998821706160804127, 11.97216163855752854798208673036, 13.06267565558064360819841292317

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