Properties

Label 2-48-1.1-c21-0-1
Degree $2$
Conductor $48$
Sign $1$
Analytic cond. $134.149$
Root an. cond. $11.5822$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.90e4·3-s − 2.12e7·5-s − 6.32e8·7-s + 3.48e9·9-s − 5.97e10·11-s + 7.38e11·13-s + 1.25e12·15-s − 8.35e12·17-s − 4.19e13·19-s + 3.73e13·21-s − 4.48e13·23-s − 2.41e13·25-s − 2.05e14·27-s − 2.76e15·29-s − 8.36e15·31-s + 3.52e15·33-s + 1.34e16·35-s − 1.77e16·37-s − 4.36e16·39-s + 1.45e17·41-s − 1.24e17·43-s − 7.41e16·45-s − 4.28e17·47-s − 1.59e17·49-s + 4.93e17·51-s − 4.77e17·53-s + 1.27e18·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.974·5-s − 0.845·7-s + 0.333·9-s − 0.694·11-s + 1.48·13-s + 0.562·15-s − 1.00·17-s − 1.56·19-s + 0.488·21-s − 0.225·23-s − 0.0505·25-s − 0.192·27-s − 1.22·29-s − 1.83·31-s + 0.401·33-s + 0.824·35-s − 0.607·37-s − 0.857·39-s + 1.69·41-s − 0.880·43-s − 0.324·45-s − 1.18·47-s − 0.284·49-s + 0.580·51-s − 0.374·53-s + 0.676·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $1$
Analytic conductor: \(134.149\)
Root analytic conductor: \(11.5822\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(0.08481030669\)
\(L(\frac12)\) \(\approx\) \(0.08481030669\)
\(L(\frac{23}{2})\) 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 \( 1 + 5.90e4T \)
good5 \( 1 + 2.12e7T + 4.76e14T^{2} \)
7 \( 1 + 6.32e8T + 5.58e17T^{2} \)
11 \( 1 + 5.97e10T + 7.40e21T^{2} \)
13 \( 1 - 7.38e11T + 2.47e23T^{2} \)
17 \( 1 + 8.35e12T + 6.90e25T^{2} \)
19 \( 1 + 4.19e13T + 7.14e26T^{2} \)
23 \( 1 + 4.48e13T + 3.94e28T^{2} \)
29 \( 1 + 2.76e15T + 5.13e30T^{2} \)
31 \( 1 + 8.36e15T + 2.08e31T^{2} \)
37 \( 1 + 1.77e16T + 8.55e32T^{2} \)
41 \( 1 - 1.45e17T + 7.38e33T^{2} \)
43 \( 1 + 1.24e17T + 2.00e34T^{2} \)
47 \( 1 + 4.28e17T + 1.30e35T^{2} \)
53 \( 1 + 4.77e17T + 1.62e36T^{2} \)
59 \( 1 + 1.61e18T + 1.54e37T^{2} \)
61 \( 1 + 3.76e18T + 3.10e37T^{2} \)
67 \( 1 - 2.81e18T + 2.22e38T^{2} \)
71 \( 1 + 1.00e19T + 7.52e38T^{2} \)
73 \( 1 + 1.72e19T + 1.34e39T^{2} \)
79 \( 1 - 3.28e19T + 7.08e39T^{2} \)
83 \( 1 + 3.05e17T + 1.99e40T^{2} \)
89 \( 1 - 2.34e20T + 8.65e40T^{2} \)
97 \( 1 + 5.92e20T + 5.27e41T^{2} \)
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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

−11.25106965839400563506751315848, −10.72259924372898671469954847623, −9.155447495857736348764367623546, −8.014248946255929123888854970691, −6.75565724177711607703954500965, −5.80802489782648943315384068513, −4.28088647509277283177005830284, −3.45474248086505151295524693937, −1.83900229815880311317218577738, −0.13082036356501804553445518374, 0.13082036356501804553445518374, 1.83900229815880311317218577738, 3.45474248086505151295524693937, 4.28088647509277283177005830284, 5.80802489782648943315384068513, 6.75565724177711607703954500965, 8.014248946255929123888854970691, 9.155447495857736348764367623546, 10.72259924372898671469954847623, 11.25106965839400563506751315848

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