Properties

Label 2-8640-1.1-c1-0-2
Degree $2$
Conductor $8640$
Sign $1$
Analytic cond. $68.9907$
Root an. cond. $8.30606$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4.60·7-s − 2.60·11-s + 0.605·13-s − 5.60·17-s − 3.60·19-s + 3·23-s + 25-s − 8.60·29-s − 1.60·31-s + 4.60·35-s − 2·37-s + 2.60·41-s − 6.60·43-s − 5.21·47-s + 14.2·49-s + 5.60·53-s + 2.60·55-s − 8.60·59-s − 10.2·61-s − 0.605·65-s − 15.2·67-s − 14.6·71-s + 5.39·73-s + 12·77-s + 4.39·79-s + 3·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.74·7-s − 0.785·11-s + 0.167·13-s − 1.35·17-s − 0.827·19-s + 0.625·23-s + 0.200·25-s − 1.59·29-s − 0.288·31-s + 0.778·35-s − 0.328·37-s + 0.406·41-s − 1.00·43-s − 0.760·47-s + 2.03·49-s + 0.769·53-s + 0.351·55-s − 1.12·59-s − 1.30·61-s − 0.0751·65-s − 1.85·67-s − 1.73·71-s + 0.631·73-s + 1.36·77-s + 0.494·79-s + 0.329·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8640\)    =    \(2^{6} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(68.9907\)
Root analytic conductor: \(8.30606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2495524965\)
\(L(\frac12)\) \(\approx\) \(0.2495524965\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good7 \( 1 + 4.60T + 7T^{2} \)
11 \( 1 + 2.60T + 11T^{2} \)
13 \( 1 - 0.605T + 13T^{2} \)
17 \( 1 + 5.60T + 17T^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + 8.60T + 29T^{2} \)
31 \( 1 + 1.60T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 + 6.60T + 43T^{2} \)
47 \( 1 + 5.21T + 47T^{2} \)
53 \( 1 - 5.60T + 53T^{2} \)
59 \( 1 + 8.60T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + 15.2T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 - 5.39T + 73T^{2} \)
79 \( 1 - 4.39T + 79T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.60256775712204749033977593540, −7.09225360929528379305254187194, −6.38218195033057287521490459607, −5.91127698403205633915644990705, −4.92971684757104726137020008901, −4.17030716142106653073355952767, −3.39799068471112309821591226189, −2.81635610571740390589594413841, −1.86506103161801755090970266778, −0.22658921592113570937878268213, 0.22658921592113570937878268213, 1.86506103161801755090970266778, 2.81635610571740390589594413841, 3.39799068471112309821591226189, 4.17030716142106653073355952767, 4.92971684757104726137020008901, 5.91127698403205633915644990705, 6.38218195033057287521490459607, 7.09225360929528379305254187194, 7.60256775712204749033977593540

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