Properties

Label 2-8016-1.1-c1-0-105
Degree $2$
Conductor $8016$
Sign $1$
Analytic cond. $64.0080$
Root an. cond. $8.00050$
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  + 3-s + 2.60·5-s + 3.58·7-s + 9-s + 4.72·11-s − 0.942·13-s + 2.60·15-s − 3.64·17-s + 2.69·19-s + 3.58·21-s + 0.381·23-s + 1.76·25-s + 27-s + 4.31·29-s − 0.400·31-s + 4.72·33-s + 9.31·35-s + 0.407·37-s − 0.942·39-s + 0.759·41-s − 4.47·43-s + 2.60·45-s − 13.2·47-s + 5.82·49-s − 3.64·51-s + 4.41·53-s + 12.2·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.16·5-s + 1.35·7-s + 0.333·9-s + 1.42·11-s − 0.261·13-s + 0.671·15-s − 0.884·17-s + 0.617·19-s + 0.781·21-s + 0.0795·23-s + 0.353·25-s + 0.192·27-s + 0.800·29-s − 0.0719·31-s + 0.822·33-s + 1.57·35-s + 0.0669·37-s − 0.150·39-s + 0.118·41-s − 0.682·43-s + 0.387·45-s − 1.92·47-s + 0.832·49-s − 0.510·51-s + 0.606·53-s + 1.65·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
Sign: $1$
Analytic conductor: \(64.0080\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.520352841\)
\(L(\frac12)\) \(\approx\) \(4.520352841\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 - 2.60T + 5T^{2} \)
7 \( 1 - 3.58T + 7T^{2} \)
11 \( 1 - 4.72T + 11T^{2} \)
13 \( 1 + 0.942T + 13T^{2} \)
17 \( 1 + 3.64T + 17T^{2} \)
19 \( 1 - 2.69T + 19T^{2} \)
23 \( 1 - 0.381T + 23T^{2} \)
29 \( 1 - 4.31T + 29T^{2} \)
31 \( 1 + 0.400T + 31T^{2} \)
37 \( 1 - 0.407T + 37T^{2} \)
41 \( 1 - 0.759T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + 13.2T + 47T^{2} \)
53 \( 1 - 4.41T + 53T^{2} \)
59 \( 1 - 5.47T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 4.38T + 67T^{2} \)
71 \( 1 - 1.17T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 8.57T + 79T^{2} \)
83 \( 1 + 0.551T + 83T^{2} \)
89 \( 1 - 0.764T + 89T^{2} \)
97 \( 1 + 4.11T + 97T^{2} \)
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

−8.040515106553373189541289125994, −6.98189068741777118209290693663, −6.62152611468076432096361288806, −5.70862502866542393286411995801, −4.97266737421881251826002294945, −4.38157248671722446130885583509, −3.50674997877368967736214908113, −2.43301935993732376572678658112, −1.79033849915491980871183302615, −1.15224419493487414603613706891, 1.15224419493487414603613706891, 1.79033849915491980871183302615, 2.43301935993732376572678658112, 3.50674997877368967736214908113, 4.38157248671722446130885583509, 4.97266737421881251826002294945, 5.70862502866542393286411995801, 6.62152611468076432096361288806, 6.98189068741777118209290693663, 8.040515106553373189541289125994

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