Properties

Label 2-8280-1.1-c1-0-25
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 − 3·7-s + 4.31·11-s − 2.31·13-s + 3.31·17-s − 1.68·19-s − 23-s + 25-s + 0.683·29-s − 9.63·31-s − 3·35-s + 11.6·37-s − 3.31·41-s + 2.63·43-s − 10.9·47-s + 2·49-s + 9.94·53-s + 4.31·55-s + 11.9·59-s + 3.68·61-s − 2.31·65-s + 2.36·67-s + 7.94·71-s + 1.68·73-s − 12.9·77-s − 5.31·83-s + 3.31·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.13·7-s + 1.30·11-s − 0.642·13-s + 0.804·17-s − 0.386·19-s − 0.208·23-s + 0.200·25-s + 0.126·29-s − 1.73·31-s − 0.507·35-s + 1.91·37-s − 0.517·41-s + 0.401·43-s − 1.59·47-s + 0.285·49-s + 1.36·53-s + 0.582·55-s + 1.55·59-s + 0.471·61-s − 0.287·65-s + 0.289·67-s + 0.943·71-s + 0.197·73-s − 1.47·77-s − 0.583·83-s + 0.359·85-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

\(L(1)\) \(\approx\) \(1.882842590\)
\(L(\frac12)\) \(\approx\) \(1.882842590\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 + 2.31T + 13T^{2} \)
17 \( 1 - 3.31T + 17T^{2} \)
19 \( 1 + 1.68T + 19T^{2} \)
29 \( 1 - 0.683T + 29T^{2} \)
31 \( 1 + 9.63T + 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 + 3.31T + 41T^{2} \)
43 \( 1 - 2.63T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 9.94T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 3.68T + 61T^{2} \)
67 \( 1 - 2.36T + 67T^{2} \)
71 \( 1 - 7.94T + 71T^{2} \)
73 \( 1 - 1.68T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 5.31T + 83T^{2} \)
89 \( 1 + 8.63T + 89T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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

−7.70983956786685106767244335223, −6.95766462884354972580795919740, −6.47622105898560459342202031306, −5.83298923175957697722262128638, −5.13531030668431488674381214115, −4.08316739956221152938840292725, −3.56441816367635607915807645338, −2.68224845031852740083856478415, −1.78540791522464592475425805967, −0.67399249946092103126652483353, 0.67399249946092103126652483353, 1.78540791522464592475425805967, 2.68224845031852740083856478415, 3.56441816367635607915807645338, 4.08316739956221152938840292725, 5.13531030668431488674381214115, 5.83298923175957697722262128638, 6.47622105898560459342202031306, 6.95766462884354972580795919740, 7.70983956786685106767244335223

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