Properties

Degree $2$
Conductor $8016$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.94·5-s + 3.19·7-s + 9-s + 0.785·11-s − 4.88·13-s − 1.94·15-s + 2.74·17-s − 1.67·19-s − 3.19·21-s + 6.23·23-s − 1.21·25-s − 27-s − 0.902·29-s − 5.13·31-s − 0.785·33-s + 6.21·35-s + 0.279·37-s + 4.88·39-s + 1.15·41-s − 4.50·43-s + 1.94·45-s + 8.29·47-s + 3.22·49-s − 2.74·51-s − 6.24·53-s + 1.52·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.869·5-s + 1.20·7-s + 0.333·9-s + 0.236·11-s − 1.35·13-s − 0.502·15-s + 0.666·17-s − 0.384·19-s − 0.697·21-s + 1.30·23-s − 0.243·25-s − 0.192·27-s − 0.167·29-s − 0.922·31-s − 0.136·33-s + 1.05·35-s + 0.0459·37-s + 0.782·39-s + 0.180·41-s − 0.687·43-s + 0.289·45-s + 1.21·47-s + 0.460·49-s − 0.384·51-s − 0.858·53-s + 0.205·55-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: $1$
Motivic weight: \(1\)
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.299811632\)
\(L(\frac12)\) \(\approx\) \(2.299811632\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.94T + 5T^{2} \)
7 \( 1 - 3.19T + 7T^{2} \)
11 \( 1 - 0.785T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 - 2.74T + 17T^{2} \)
19 \( 1 + 1.67T + 19T^{2} \)
23 \( 1 - 6.23T + 23T^{2} \)
29 \( 1 + 0.902T + 29T^{2} \)
31 \( 1 + 5.13T + 31T^{2} \)
37 \( 1 - 0.279T + 37T^{2} \)
41 \( 1 - 1.15T + 41T^{2} \)
43 \( 1 + 4.50T + 43T^{2} \)
47 \( 1 - 8.29T + 47T^{2} \)
53 \( 1 + 6.24T + 53T^{2} \)
59 \( 1 - 9.18T + 59T^{2} \)
61 \( 1 - 0.717T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 - 2.50T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 4.93T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 1.53T + 97T^{2} \)
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

−7.75077282465788190439582106846, −7.11201565444419446678736040955, −6.47015239431818098749578828188, −5.45886358573859670096875546339, −5.23688935479547136116978659706, −4.55564645297523329743496884835, −3.57903204346494553360185194012, −2.37716355209986162408921028392, −1.81068800750492243838849535684, −0.792429339941357491478249837240, 0.792429339941357491478249837240, 1.81068800750492243838849535684, 2.37716355209986162408921028392, 3.57903204346494553360185194012, 4.55564645297523329743496884835, 5.23688935479547136116978659706, 5.45886358573859670096875546339, 6.47015239431818098749578828188, 7.11201565444419446678736040955, 7.75077282465788190439582106846

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