Properties

Label 2-48-1.1-c13-0-10
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 729·3-s − 3.02e4·5-s − 2.35e5·7-s + 5.31e5·9-s + 1.11e7·11-s + 8.04e6·13-s − 2.20e7·15-s − 1.17e8·17-s + 2.14e8·19-s − 1.71e8·21-s − 8.30e8·23-s − 3.08e8·25-s + 3.87e8·27-s − 1.25e9·29-s − 6.15e9·31-s + 8.15e9·33-s + 7.10e9·35-s − 5.49e9·37-s + 5.86e9·39-s − 4.67e9·41-s − 7.11e9·43-s − 1.60e10·45-s + 2.95e10·47-s − 4.16e10·49-s − 8.56e10·51-s − 2.04e11·53-s − 3.37e11·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.864·5-s − 0.755·7-s + 1/3·9-s + 1.90·11-s + 0.462·13-s − 0.499·15-s − 1.18·17-s + 1.04·19-s − 0.436·21-s − 1.16·23-s − 0.252·25-s + 0.192·27-s − 0.390·29-s − 1.24·31-s + 1.09·33-s + 0.653·35-s − 0.352·37-s + 0.267·39-s − 0.153·41-s − 0.171·43-s − 0.288·45-s + 0.399·47-s − 0.429·49-s − 0.681·51-s − 1.26·53-s − 1.64·55-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 \( 1 + 6042 p T + p^{13} T^{2} \)
7 \( 1 + 33584 p T + p^{13} T^{2} \)
11 \( 1 - 1016628 p T + p^{13} T^{2} \)
13 \( 1 - 8049614 T + p^{13} T^{2} \)
17 \( 1 + 117494622 T + p^{13} T^{2} \)
19 \( 1 - 214061380 T + p^{13} T^{2} \)
23 \( 1 + 830555544 T + p^{13} T^{2} \)
29 \( 1 + 1252400250 T + p^{13} T^{2} \)
31 \( 1 + 6159350552 T + p^{13} T^{2} \)
37 \( 1 + 5498191402 T + p^{13} T^{2} \)
41 \( 1 + 4678687878 T + p^{13} T^{2} \)
43 \( 1 + 7115013764 T + p^{13} T^{2} \)
47 \( 1 - 29528776992 T + p^{13} T^{2} \)
53 \( 1 + 204125042466 T + p^{13} T^{2} \)
59 \( 1 - 29909821020 T + p^{13} T^{2} \)
61 \( 1 + 134392006738 T + p^{13} T^{2} \)
67 \( 1 + 348518801948 T + p^{13} T^{2} \)
71 \( 1 + 1314335409192 T + p^{13} T^{2} \)
73 \( 1 + 1178875922326 T + p^{13} T^{2} \)
79 \( 1 - 1072420659640 T + p^{13} T^{2} \)
83 \( 1 + 1124025139644 T + p^{13} T^{2} \)
89 \( 1 - 2235610909530 T + p^{13} T^{2} \)
97 \( 1 + 14215257165502 T + p^{13} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.19920204719357331514546506189, −11.28874863133100751882322525350, −9.612401555966694054865613938696, −8.766689992903158198288669392222, −7.37366932865826266223476660431, −6.26216009026594827617838618341, −4.14198427325835632091138007571, −3.41772842943156544165541984449, −1.60547776928845228034986702729, 0, 1.60547776928845228034986702729, 3.41772842943156544165541984449, 4.14198427325835632091138007571, 6.26216009026594827617838618341, 7.37366932865826266223476660431, 8.766689992903158198288669392222, 9.612401555966694054865613938696, 11.28874863133100751882322525350, 12.19920204719357331514546506189

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