Properties

Degree $2$
Conductor $2016$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 7-s − 2·11-s − 2·13-s + 4·19-s − 6·23-s + 11·25-s + 10·29-s + 8·31-s − 4·35-s + 10·37-s + 4·41-s + 8·43-s − 4·47-s + 49-s − 10·53-s − 8·55-s + 8·59-s − 6·61-s − 8·65-s − 4·67-s + 14·71-s + 6·73-s + 2·77-s − 4·79-s − 12·83-s − 4·89-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.377·7-s − 0.603·11-s − 0.554·13-s + 0.917·19-s − 1.25·23-s + 11/5·25-s + 1.85·29-s + 1.43·31-s − 0.676·35-s + 1.64·37-s + 0.624·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s − 1.37·53-s − 1.07·55-s + 1.04·59-s − 0.768·61-s − 0.992·65-s − 0.488·67-s + 1.66·71-s + 0.702·73-s + 0.227·77-s − 0.450·79-s − 1.31·83-s − 0.423·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.354652703\)
\(L(\frac12)\) \(\approx\) \(2.354652703\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−9.481276071511481565581284791220, −8.409578506072020402089900906100, −7.61971828815925419087584402587, −6.49198050199370736249656456104, −6.07899407066809115934805106887, −5.24330200469260993508502492214, −4.44802041941911311166014127039, −2.86972543343592766466945454261, −2.39751668918426191936188305346, −1.07233750136519077963315820080, 1.07233750136519077963315820080, 2.39751668918426191936188305346, 2.86972543343592766466945454261, 4.44802041941911311166014127039, 5.24330200469260993508502492214, 6.07899407066809115934805106887, 6.49198050199370736249656456104, 7.61971828815925419087584402587, 8.409578506072020402089900906100, 9.481276071511481565581284791220

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