Properties

Label 2-48-1.1-c13-0-11
Degree $2$
Conductor $48$
Sign $-1$
Analytic cond. $51.4708$
Root an. cond. $7.17431$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 729·3-s − 1.48e4·5-s + 6.28e4·7-s + 5.31e5·9-s − 5.10e6·11-s + 1.14e7·13-s − 1.08e7·15-s + 1.19e8·17-s − 3.32e8·19-s + 4.58e7·21-s − 3.50e8·23-s − 1.00e9·25-s + 3.87e8·27-s − 1.76e9·29-s + 3.93e9·31-s − 3.72e9·33-s − 9.34e8·35-s − 7.80e9·37-s + 8.37e9·39-s + 5.28e10·41-s − 2.60e10·43-s − 7.89e9·45-s − 1.42e11·47-s − 9.29e10·49-s + 8.74e10·51-s + 1.37e10·53-s + 7.58e10·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.425·5-s + 0.202·7-s + 1/3·9-s − 0.868·11-s + 0.659·13-s − 0.245·15-s + 1.20·17-s − 1.62·19-s + 0.116·21-s − 0.494·23-s − 0.819·25-s + 0.192·27-s − 0.549·29-s + 0.796·31-s − 0.501·33-s − 0.0858·35-s − 0.500·37-s + 0.380·39-s + 1.73·41-s − 0.627·43-s − 0.141·45-s − 1.92·47-s − 0.959·49-s + 0.695·51-s + 0.0853·53-s + 0.369·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-1$
Analytic conductor: \(51.4708\)
Root analytic conductor: \(7.17431\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 48,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) 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 \( 1 - p^{6} T \)
good5 \( 1 + 594 p^{2} T + p^{13} T^{2} \)
7 \( 1 - 62896 T + p^{13} T^{2} \)
11 \( 1 + 464076 p T + p^{13} T^{2} \)
13 \( 1 - 11484110 T + p^{13} T^{2} \)
17 \( 1 - 119964834 T + p^{13} T^{2} \)
19 \( 1 + 332601020 T + p^{13} T^{2} \)
23 \( 1 + 350924184 T + p^{13} T^{2} \)
29 \( 1 + 1761101946 T + p^{13} T^{2} \)
31 \( 1 - 3934224616 T + p^{13} T^{2} \)
37 \( 1 + 210907234 p T + p^{13} T^{2} \)
41 \( 1 - 52882647930 T + p^{13} T^{2} \)
43 \( 1 + 26018412164 T + p^{13} T^{2} \)
47 \( 1 + 142370739936 T + p^{13} T^{2} \)
53 \( 1 - 13770034398 T + p^{13} T^{2} \)
59 \( 1 + 336464984484 T + p^{13} T^{2} \)
61 \( 1 + 677260793938 T + p^{13} T^{2} \)
67 \( 1 + 262301598236 T + p^{13} T^{2} \)
71 \( 1 + 1594961300520 T + p^{13} T^{2} \)
73 \( 1 - 578812819562 T + p^{13} T^{2} \)
79 \( 1 + 2495818789448 T + p^{13} T^{2} \)
83 \( 1 - 2693235578436 T + p^{13} T^{2} \)
89 \( 1 + 7935538832550 T + p^{13} T^{2} \)
97 \( 1 + 7858601662 T + p^{13} T^{2} \)
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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

−12.39454109129735506169261565370, −11.03098341978973556413686393957, −9.903179885823731068204446726840, −8.419499589405583573137361232674, −7.68771504055974879779442176270, −6.03392003828959394487577222877, −4.42461377420497324337781825620, −3.15626178334346068027156895474, −1.70576147122579746168589844033, 0, 1.70576147122579746168589844033, 3.15626178334346068027156895474, 4.42461377420497324337781825620, 6.03392003828959394487577222877, 7.68771504055974879779442176270, 8.419499589405583573137361232674, 9.903179885823731068204446726840, 11.03098341978973556413686393957, 12.39454109129735506169261565370

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