Properties

Label 2-3960-1.1-c1-0-7
Degree $2$
Conductor $3960$
Sign $1$
Analytic cond. $31.6207$
Root an. cond. $5.62323$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s + 11-s − 6·13-s − 3·17-s − 5·19-s + 2·23-s + 25-s + 5·29-s + 5·31-s − 35-s − 37-s + 2·41-s + 12·43-s + 2·47-s − 6·49-s + 13·53-s − 55-s − 2·59-s + 61-s + 6·65-s + 16·67-s − 15·71-s + 10·73-s + 77-s + 2·79-s + 14·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.301·11-s − 1.66·13-s − 0.727·17-s − 1.14·19-s + 0.417·23-s + 1/5·25-s + 0.928·29-s + 0.898·31-s − 0.169·35-s − 0.164·37-s + 0.312·41-s + 1.82·43-s + 0.291·47-s − 6/7·49-s + 1.78·53-s − 0.134·55-s − 0.260·59-s + 0.128·61-s + 0.744·65-s + 1.95·67-s − 1.78·71-s + 1.17·73-s + 0.113·77-s + 0.225·79-s + 1.53·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(31.6207\)
Root analytic conductor: \(5.62323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.440004295\)
\(L(\frac12)\) \(\approx\) \(1.440004295\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 - T \)
good7 \( 1 - T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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

−8.438674817423357570936470035554, −7.73433059133812301192977364984, −6.99970383505905243647391751086, −6.42321132891736916212149036139, −5.34122428081478900562427414965, −4.56735255286468441470482194742, −4.09982993045677391676142437218, −2.78789847884678517090591066257, −2.14467665127329154435692730090, −0.67084152583760764906013623104, 0.67084152583760764906013623104, 2.14467665127329154435692730090, 2.78789847884678517090591066257, 4.09982993045677391676142437218, 4.56735255286468441470482194742, 5.34122428081478900562427414965, 6.42321132891736916212149036139, 6.99970383505905243647391751086, 7.73433059133812301192977364984, 8.438674817423357570936470035554

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