Properties

Label 2-8280-1.1-c1-0-56
Degree $2$
Conductor $8280$
Sign $1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
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  − 5-s + 5.12·7-s + 4·11-s − 0.561·13-s + 3.12·17-s + 4·19-s − 23-s + 25-s + 8.56·29-s + 1.43·31-s − 5.12·35-s − 7.12·37-s − 0.561·41-s − 9.12·43-s + 3.68·47-s + 19.2·49-s + 4.24·53-s − 4·55-s + 6.24·59-s + 11.1·61-s + 0.561·65-s + 6.24·67-s − 3.68·71-s + 16.5·73-s + 20.4·77-s − 10.2·79-s − 12·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.93·7-s + 1.20·11-s − 0.155·13-s + 0.757·17-s + 0.917·19-s − 0.208·23-s + 0.200·25-s + 1.58·29-s + 0.258·31-s − 0.865·35-s − 1.17·37-s − 0.0876·41-s − 1.39·43-s + 0.537·47-s + 2.74·49-s + 0.583·53-s − 0.539·55-s + 0.813·59-s + 1.42·61-s + 0.0696·65-s + 0.763·67-s − 0.437·71-s + 1.93·73-s + 2.33·77-s − 1.15·79-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.071876018\)
\(L(\frac12)\) \(\approx\) \(3.071876018\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
23 \( 1 + T \)
good7 \( 1 - 5.12T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 0.561T + 13T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 - 8.56T + 29T^{2} \)
31 \( 1 - 1.43T + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 + 0.561T + 41T^{2} \)
43 \( 1 + 9.12T + 43T^{2} \)
47 \( 1 - 3.68T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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

−7.923769643146157676178375968490, −7.14489581671411361885464710153, −6.62634641992374752768862936009, −5.42761181959295018180435065252, −5.11083247686522509221237981106, −4.24755609493233515197054795986, −3.68936845469457780060311823362, −2.60999853986396129144298821915, −1.53459503631254466209190248204, −0.991002153158277674679225593400, 0.991002153158277674679225593400, 1.53459503631254466209190248204, 2.60999853986396129144298821915, 3.68936845469457780060311823362, 4.24755609493233515197054795986, 5.11083247686522509221237981106, 5.42761181959295018180435065252, 6.62634641992374752768862936009, 7.14489581671411361885464710153, 7.923769643146157676178375968490

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