Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
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 − 1.38·5-s + 0.260·7-s + 9-s + 3.57·11-s − 1.29·13-s − 1.38·15-s − 4.24·17-s − 5.37·19-s + 0.260·21-s − 5.60·23-s − 3.09·25-s + 27-s + 2.23·29-s + 9.62·31-s + 3.57·33-s − 0.359·35-s + 0.0808·37-s − 1.29·39-s + 4.29·41-s + 10.5·43-s − 1.38·45-s + 6.20·47-s − 6.93·49-s − 4.24·51-s + 12.3·53-s − 4.94·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.617·5-s + 0.0983·7-s + 0.333·9-s + 1.07·11-s − 0.360·13-s − 0.356·15-s − 1.02·17-s − 1.23·19-s + 0.0567·21-s − 1.16·23-s − 0.618·25-s + 0.192·27-s + 0.414·29-s + 1.72·31-s + 0.623·33-s − 0.0607·35-s + 0.0132·37-s − 0.208·39-s + 0.670·41-s + 1.60·43-s − 0.205·45-s + 0.904·47-s − 0.990·49-s − 0.594·51-s + 1.70·53-s − 0.666·55-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

\( d \)  =  \(2\)
\( N \)  =  \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8016} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8016,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.047704633\)
\(L(\frac12)\)  \(\approx\)  \(2.047704633\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 + 1.38T + 5T^{2} \)
7 \( 1 - 0.260T + 7T^{2} \)
11 \( 1 - 3.57T + 11T^{2} \)
13 \( 1 + 1.29T + 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 + 5.37T + 19T^{2} \)
23 \( 1 + 5.60T + 23T^{2} \)
29 \( 1 - 2.23T + 29T^{2} \)
31 \( 1 - 9.62T + 31T^{2} \)
37 \( 1 - 0.0808T + 37T^{2} \)
41 \( 1 - 4.29T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 6.20T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 + 3.18T + 59T^{2} \)
61 \( 1 + 6.40T + 61T^{2} \)
67 \( 1 + 2.58T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 + 4.33T + 83T^{2} \)
89 \( 1 - 8.61T + 89T^{2} \)
97 \( 1 - 0.0718T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.83137979426925623593141743535, −7.29539827702334342426388603560, −6.36732505607228350571924530634, −6.07109262975427411841818545261, −4.62872170554824814966831850604, −4.27890751159020775949609651302, −3.68095895234707889972729924722, −2.56411283094595966510990144567, −1.95837277679158386426423025315, −0.67631928690572455228011411187, 0.67631928690572455228011411187, 1.95837277679158386426423025315, 2.56411283094595966510990144567, 3.68095895234707889972729924722, 4.27890751159020775949609651302, 4.62872170554824814966831850604, 6.07109262975427411841818545261, 6.36732505607228350571924530634, 7.29539827702334342426388603560, 7.83137979426925623593141743535

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