Properties

Label 2-7280-1.1-c1-0-62
Degree $2$
Conductor $7280$
Sign $1$
Analytic cond. $58.1310$
Root an. cond. $7.62437$
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  + 2.30·3-s + 5-s − 7-s + 2.30·9-s − 2.60·11-s − 13-s + 2.30·15-s + 2.69·17-s + 2.69·19-s − 2.30·21-s + 8·23-s + 25-s − 1.60·27-s − 8.90·29-s + 4.30·31-s − 6·33-s − 35-s + 6.90·37-s − 2.30·39-s − 0.697·41-s + 2·43-s + 2.30·45-s − 2.60·47-s + 49-s + 6.21·51-s + 4.60·53-s − 2.60·55-s + ⋯
L(s)  = 1  + 1.32·3-s + 0.447·5-s − 0.377·7-s + 0.767·9-s − 0.785·11-s − 0.277·13-s + 0.594·15-s + 0.654·17-s + 0.618·19-s − 0.502·21-s + 1.66·23-s + 0.200·25-s − 0.308·27-s − 1.65·29-s + 0.772·31-s − 1.04·33-s − 0.169·35-s + 1.13·37-s − 0.368·39-s − 0.108·41-s + 0.304·43-s + 0.343·45-s − 0.380·47-s + 0.142·49-s + 0.869·51-s + 0.632·53-s − 0.351·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7280\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(58.1310\)
Root analytic conductor: \(7.62437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.493620425\)
\(L(\frac12)\) \(\approx\) \(3.493620425\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 + T \)
good3 \( 1 - 2.30T + 3T^{2} \)
11 \( 1 + 2.60T + 11T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 - 2.69T + 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + 8.90T + 29T^{2} \)
31 \( 1 - 4.30T + 31T^{2} \)
37 \( 1 - 6.90T + 37T^{2} \)
41 \( 1 + 0.697T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 2.60T + 47T^{2} \)
53 \( 1 - 4.60T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 - 1.90T + 67T^{2} \)
71 \( 1 + 4.60T + 71T^{2} \)
73 \( 1 + 1.39T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + 9.81T + 83T^{2} \)
89 \( 1 + 3.69T + 89T^{2} \)
97 \( 1 - 1.21T + 97T^{2} \)
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

−7.940592565118556929540042495225, −7.36297738989938860766192743960, −6.75375457468286213414776593508, −5.62333922567089466448710665280, −5.24016072055114202110069411429, −4.13411806619104256058926754558, −3.30440747840295129030994366927, −2.76352627737758405324713883851, −2.09382477850061284249455356645, −0.885987301316753567224652147534, 0.885987301316753567224652147534, 2.09382477850061284249455356645, 2.76352627737758405324713883851, 3.30440747840295129030994366927, 4.13411806619104256058926754558, 5.24016072055114202110069411429, 5.62333922567089466448710665280, 6.75375457468286213414776593508, 7.36297738989938860766192743960, 7.940592565118556929540042495225

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