Properties

Label 2-48-1.1-c23-0-18
Degree $2$
Conductor $48$
Sign $-1$
Analytic cond. $160.897$
Root an. cond. $12.6845$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.77e5·3-s − 4.08e5·5-s + 9.39e9·7-s + 3.13e10·9-s − 8.83e11·11-s − 2.21e12·13-s + 7.23e10·15-s + 2.27e14·17-s + 6.53e13·19-s − 1.66e15·21-s − 5.34e15·23-s − 1.19e16·25-s − 5.55e15·27-s − 4.32e16·29-s − 1.98e17·31-s + 1.56e17·33-s − 3.83e15·35-s + 1.09e17·37-s + 3.91e17·39-s + 6.10e18·41-s − 2.13e18·43-s − 1.28e16·45-s − 1.58e19·47-s + 6.09e19·49-s − 4.03e19·51-s + 6.82e18·53-s + 3.60e17·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.00373·5-s + 1.79·7-s + 0.333·9-s − 0.933·11-s − 0.342·13-s + 0.00215·15-s + 1.61·17-s + 0.128·19-s − 1.03·21-s − 1.16·23-s − 0.999·25-s − 0.192·27-s − 0.658·29-s − 1.40·31-s + 0.539·33-s − 0.00671·35-s + 0.101·37-s + 0.197·39-s + 1.73·41-s − 0.349·43-s − 0.00124·45-s − 0.935·47-s + 2.22·49-s − 0.931·51-s + 0.101·53-s + 0.00349·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-1$
Analytic conductor: \(160.897\)
Root analytic conductor: \(12.6845\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 48,\ (\ :23/2),\ -1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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 \( 1 + 1.77e5T \)
good5 \( 1 + 4.08e5T + 1.19e16T^{2} \)
7 \( 1 - 9.39e9T + 2.73e19T^{2} \)
11 \( 1 + 8.83e11T + 8.95e23T^{2} \)
13 \( 1 + 2.21e12T + 4.17e25T^{2} \)
17 \( 1 - 2.27e14T + 1.99e28T^{2} \)
19 \( 1 - 6.53e13T + 2.57e29T^{2} \)
23 \( 1 + 5.34e15T + 2.08e31T^{2} \)
29 \( 1 + 4.32e16T + 4.31e33T^{2} \)
31 \( 1 + 1.98e17T + 2.00e34T^{2} \)
37 \( 1 - 1.09e17T + 1.17e36T^{2} \)
41 \( 1 - 6.10e18T + 1.24e37T^{2} \)
43 \( 1 + 2.13e18T + 3.71e37T^{2} \)
47 \( 1 + 1.58e19T + 2.87e38T^{2} \)
53 \( 1 - 6.82e18T + 4.55e39T^{2} \)
59 \( 1 + 2.82e20T + 5.36e40T^{2} \)
61 \( 1 - 2.11e20T + 1.15e41T^{2} \)
67 \( 1 - 1.00e21T + 9.99e41T^{2} \)
71 \( 1 - 2.75e21T + 3.79e42T^{2} \)
73 \( 1 + 4.77e20T + 7.18e42T^{2} \)
79 \( 1 - 5.85e21T + 4.42e43T^{2} \)
83 \( 1 + 2.00e22T + 1.37e44T^{2} \)
89 \( 1 + 1.15e22T + 6.85e44T^{2} \)
97 \( 1 - 9.73e22T + 4.96e45T^{2} \)
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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

−10.81208224408548791797351949279, −9.714887635398139284310751516200, −7.998651354983664488642441834936, −7.59013713145671304485773152779, −5.70491188449638427267641030257, −5.11932667355313533777161189333, −3.90064361836643802895883080759, −2.20218897228025168740722839432, −1.28537192455792517765599342808, 0, 1.28537192455792517765599342808, 2.20218897228025168740722839432, 3.90064361836643802895883080759, 5.11932667355313533777161189333, 5.70491188449638427267641030257, 7.59013713145671304485773152779, 7.998651354983664488642441834936, 9.714887635398139284310751516200, 10.81208224408548791797351949279

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