Properties

Label 2-7728-1.1-c1-0-17
Degree $2$
Conductor $7728$
Sign $1$
Analytic cond. $61.7083$
Root an. cond. $7.85546$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.16·5-s + 7-s + 9-s − 6.56·11-s − 1.77·13-s − 1.16·15-s + 4.94·17-s − 1.64·19-s − 21-s − 23-s − 3.64·25-s − 27-s − 0.671·29-s − 2.94·31-s + 6.56·33-s + 1.16·35-s − 0.671·37-s + 1.77·39-s + 3.91·41-s + 10.3·43-s + 1.16·45-s − 8.81·47-s + 49-s − 4.94·51-s + 5.05·53-s − 7.64·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.520·5-s + 0.377·7-s + 0.333·9-s − 1.97·11-s − 0.493·13-s − 0.300·15-s + 1.19·17-s − 0.377·19-s − 0.218·21-s − 0.208·23-s − 0.728·25-s − 0.192·27-s − 0.124·29-s − 0.528·31-s + 1.14·33-s + 0.196·35-s − 0.110·37-s + 0.284·39-s + 0.611·41-s + 1.57·43-s + 0.173·45-s − 1.28·47-s + 0.142·49-s − 0.692·51-s + 0.694·53-s − 1.03·55-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: no
Arithmetic: yes
Character: $\chi_{7728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.301595537\)
\(L(\frac12)\) \(\approx\) \(1.301595537\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.16T + 5T^{2} \)
11 \( 1 + 6.56T + 11T^{2} \)
13 \( 1 + 1.77T + 13T^{2} \)
17 \( 1 - 4.94T + 17T^{2} \)
19 \( 1 + 1.64T + 19T^{2} \)
29 \( 1 + 0.671T + 29T^{2} \)
31 \( 1 + 2.94T + 31T^{2} \)
37 \( 1 + 0.671T + 37T^{2} \)
41 \( 1 - 3.91T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 8.81T + 47T^{2} \)
53 \( 1 - 5.05T + 53T^{2} \)
59 \( 1 - 3.05T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 6.98T + 67T^{2} \)
71 \( 1 - 9.62T + 71T^{2} \)
73 \( 1 - 9.47T + 73T^{2} \)
79 \( 1 - 9.48T + 79T^{2} \)
83 \( 1 - 0.614T + 83T^{2} \)
89 \( 1 + 1.08T + 89T^{2} \)
97 \( 1 - 0.463T + 97T^{2} \)
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

−7.63600096713859772368976002528, −7.46464050248327204287230613800, −6.26212390464719376970623684520, −5.67367978127690572224310577505, −5.20178010440599914575947186589, −4.56009555060597325119929796750, −3.49318939230459470393832313341, −2.53357074697566990631712485802, −1.86453261746021599497412808038, −0.56794689726355308826718346030, 0.56794689726355308826718346030, 1.86453261746021599497412808038, 2.53357074697566990631712485802, 3.49318939230459470393832313341, 4.56009555060597325119929796750, 5.20178010440599914575947186589, 5.67367978127690572224310577505, 6.26212390464719376970623684520, 7.46464050248327204287230613800, 7.63600096713859772368976002528

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