Properties

Label 2-48-1.1-c21-0-11
Degree $2$
Conductor $48$
Sign $-1$
Analytic cond. $134.149$
Root an. cond. $11.5822$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.90e4·3-s − 4.15e7·5-s − 5.38e8·7-s + 3.48e9·9-s + 6.41e10·11-s − 1.30e11·13-s − 2.45e12·15-s + 8.24e12·17-s − 1.34e13·19-s − 3.17e13·21-s + 2.33e14·23-s + 1.24e15·25-s + 2.05e14·27-s − 2.02e15·29-s + 6.86e15·31-s + 3.78e15·33-s + 2.23e16·35-s + 3.44e15·37-s − 7.73e15·39-s − 2.18e16·41-s + 7.17e16·43-s − 1.44e17·45-s − 2.83e17·47-s − 2.68e17·49-s + 4.86e17·51-s − 2.17e18·53-s − 2.66e18·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.90·5-s − 0.720·7-s + 1/3·9-s + 0.745·11-s − 0.263·13-s − 1.09·15-s + 0.991·17-s − 0.504·19-s − 0.415·21-s + 1.17·23-s + 2.61·25-s + 0.192·27-s − 0.893·29-s + 1.50·31-s + 0.430·33-s + 1.36·35-s + 0.117·37-s − 0.152·39-s − 0.254·41-s + 0.506·43-s − 0.633·45-s − 0.786·47-s − 0.480·49-s + 0.572·51-s − 1.70·53-s − 1.41·55-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 48,\ (\ :21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) 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 \( 1 - p^{10} T \)
good5 \( 1 + 8302554 p T + p^{21} T^{2} \)
7 \( 1 + 76918544 p T + p^{21} T^{2} \)
11 \( 1 - 64113040188 T + p^{21} T^{2} \)
13 \( 1 + 10075392922 p T + p^{21} T^{2} \)
17 \( 1 - 8242029723618 T + p^{21} T^{2} \)
19 \( 1 + 710110618580 p T + p^{21} T^{2} \)
23 \( 1 - 233184825844776 T + p^{21} T^{2} \)
29 \( 1 + 2024562031123770 T + p^{21} T^{2} \)
31 \( 1 - 6869194988701768 T + p^{21} T^{2} \)
37 \( 1 - 3443998107027638 T + p^{21} T^{2} \)
41 \( 1 + 21842403084625158 T + p^{21} T^{2} \)
43 \( 1 - 71792816814133756 T + p^{21} T^{2} \)
47 \( 1 + 283544719418655648 T + p^{21} T^{2} \)
53 \( 1 + 2172285419049898146 T + p^{21} T^{2} \)
59 \( 1 + 1534831476719068260 T + p^{21} T^{2} \)
61 \( 1 - 4311589520797626062 T + p^{21} T^{2} \)
67 \( 1 + 9243910904037307868 T + p^{21} T^{2} \)
71 \( 1 - 20387361256404760728 T + p^{21} T^{2} \)
73 \( 1 - 16617754439328636074 T + p^{21} T^{2} \)
79 \( 1 + 67940304745507627880 T + p^{21} T^{2} \)
83 \( 1 + 39503732340682314684 T + p^{21} T^{2} \)
89 \( 1 - 41611676186839694490 T + p^{21} T^{2} \)
97 \( 1 - 57181473208903260098 T + p^{21} T^{2} \)
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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

−11.07738234589460099091851896590, −9.630269040377818630510695736795, −8.481097729202020181264965283380, −7.57940453816250549477265515073, −6.61008811308252124947048281132, −4.67762357713765982851322356213, −3.65863748091556832753967185921, −2.97521147694061090311581284500, −1.09867075639279792466623475976, 0, 1.09867075639279792466623475976, 2.97521147694061090311581284500, 3.65863748091556832753967185921, 4.67762357713765982851322356213, 6.61008811308252124947048281132, 7.57940453816250549477265515073, 8.481097729202020181264965283380, 9.630269040377818630510695736795, 11.07738234589460099091851896590

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