Properties

Label 2-7728-1.1-c1-0-108
Degree $2$
Conductor $7728$
Sign $-1$
Analytic cond. $61.7083$
Root an. cond. $7.85546$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 7-s + 9-s + 4·13-s − 2·15-s − 6·19-s + 21-s + 23-s − 25-s + 27-s − 6·29-s + 10·31-s − 2·35-s − 6·37-s + 4·39-s − 2·41-s − 12·43-s − 2·45-s − 10·47-s + 49-s − 10·53-s − 6·57-s + 12·59-s − 14·61-s + 63-s − 8·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 0.516·15-s − 1.37·19-s + 0.218·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.79·31-s − 0.338·35-s − 0.986·37-s + 0.640·39-s − 0.312·41-s − 1.82·43-s − 0.298·45-s − 1.45·47-s + 1/7·49-s − 1.37·53-s − 0.794·57-s + 1.56·59-s − 1.79·61-s + 0.125·63-s − 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7728\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(61.7083\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−7.79907827245742740653272644248, −6.71026616120460880523468474478, −6.44000461604244278091207509392, −5.26181540214267768138253241203, −4.55945824392917822436280024408, −3.77787025572136786062189887460, −3.35389678449645947910890096015, −2.21104651419759159123486445031, −1.37350775012907994869962001667, 0, 1.37350775012907994869962001667, 2.21104651419759159123486445031, 3.35389678449645947910890096015, 3.77787025572136786062189887460, 4.55945824392917822436280024408, 5.26181540214267768138253241203, 6.44000461604244278091207509392, 6.71026616120460880523468474478, 7.79907827245742740653272644248

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