Properties

Label 2-7728-1.1-c1-0-105
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 + 3·5-s + 7-s + 9-s − 4·11-s − 3·13-s − 3·15-s − 4·17-s − 21-s − 23-s + 4·25-s − 27-s + 3·29-s + 6·31-s + 4·33-s + 3·35-s − 9·37-s + 3·39-s + 9·41-s + 3·43-s + 3·45-s + 7·47-s + 49-s + 4·51-s − 4·53-s − 12·55-s − 6·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.832·13-s − 0.774·15-s − 0.970·17-s − 0.218·21-s − 0.208·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s + 1.07·31-s + 0.696·33-s + 0.507·35-s − 1.47·37-s + 0.480·39-s + 1.40·41-s + 0.457·43-s + 0.447·45-s + 1.02·47-s + 1/7·49-s + 0.560·51-s − 0.549·53-s − 1.61·55-s − 0.781·59-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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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

−7.39131206325256446041809734207, −6.76424663937870738066321018325, −5.99234362564232113795365538158, −5.45713141654302121882301471663, −4.87107523373367854031150290509, −4.20290822415985263475630282402, −2.70777129824037519230002306274, −2.34259324699707868728412627679, −1.33003807742036272405319904717, 0, 1.33003807742036272405319904717, 2.34259324699707868728412627679, 2.70777129824037519230002306274, 4.20290822415985263475630282402, 4.87107523373367854031150290509, 5.45713141654302121882301471663, 5.99234362564232113795365538158, 6.76424663937870738066321018325, 7.39131206325256446041809734207

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