Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
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  + 1.11·2-s − 3-s − 0.753·4-s + 0.900·5-s − 1.11·6-s − 3.07·8-s + 9-s + 1.00·10-s − 4.69·11-s + 0.753·12-s − 1.15·13-s − 0.900·15-s − 1.92·16-s − 5.04·17-s + 1.11·18-s − 0.849·19-s − 0.678·20-s − 5.24·22-s − 1.63·23-s + 3.07·24-s − 4.18·25-s − 1.29·26-s − 27-s + 7.97·29-s − 1.00·30-s − 2.94·31-s + 3.99·32-s + ⋯
L(s)  = 1  + 0.789·2-s − 0.577·3-s − 0.376·4-s + 0.402·5-s − 0.455·6-s − 1.08·8-s + 0.333·9-s + 0.317·10-s − 1.41·11-s + 0.217·12-s − 0.320·13-s − 0.232·15-s − 0.481·16-s − 1.22·17-s + 0.263·18-s − 0.194·19-s − 0.151·20-s − 1.11·22-s − 0.341·23-s + 0.627·24-s − 0.837·25-s − 0.253·26-s − 0.192·27-s + 1.48·29-s − 0.183·30-s − 0.528·31-s + 0.706·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.116774173$
$L(\frac12)$  $\approx$  $1.116774173$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 1.11T + 2T^{2} \)
5 \( 1 - 0.900T + 5T^{2} \)
11 \( 1 + 4.69T + 11T^{2} \)
13 \( 1 + 1.15T + 13T^{2} \)
17 \( 1 + 5.04T + 17T^{2} \)
19 \( 1 + 0.849T + 19T^{2} \)
23 \( 1 + 1.63T + 23T^{2} \)
29 \( 1 - 7.97T + 29T^{2} \)
31 \( 1 + 2.94T + 31T^{2} \)
37 \( 1 - 3.09T + 37T^{2} \)
43 \( 1 + 3.01T + 43T^{2} \)
47 \( 1 - 8.14T + 47T^{2} \)
53 \( 1 + 8.01T + 53T^{2} \)
59 \( 1 - 2.60T + 59T^{2} \)
61 \( 1 - 7.31T + 61T^{2} \)
67 \( 1 - 0.796T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 1.76T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 0.878T + 97T^{2} \)
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

−8.117037039313578229608608347935, −7.20397609225205451251746749072, −6.37279823184760832319959866240, −5.82208994906381510795846508520, −5.12072618278200035696316882186, −4.62959065489049260941622264139, −3.88105550847835502547606768572, −2.79148899664193457146832821037, −2.11875390968889602336906975717, −0.48075617908203773384736026197, 0.48075617908203773384736026197, 2.11875390968889602336906975717, 2.79148899664193457146832821037, 3.88105550847835502547606768572, 4.62959065489049260941622264139, 5.12072618278200035696316882186, 5.82208994906381510795846508520, 6.37279823184760832319959866240, 7.20397609225205451251746749072, 8.117037039313578229608608347935

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