Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s − 3-s − 3.41·5-s + 1.41·6-s + 2.82·8-s + 9-s + 4.82·10-s + 2.41·11-s − 6.24·13-s + 3.41·15-s − 4.00·16-s + 0.414·17-s − 1.41·18-s + 5.41·19-s − 3.41·22-s − 1.41·23-s − 2.82·24-s + 6.65·25-s + 8.82·26-s − 27-s − 6.07·29-s − 4.82·30-s + 3·31-s − 2.41·33-s − 0.585·34-s + ⋯
L(s)  = 1  − 1.00·2-s − 0.577·3-s − 1.52·5-s + 0.577·6-s + 0.999·8-s + 0.333·9-s + 1.52·10-s + 0.727·11-s − 1.73·13-s + 0.881·15-s − 1.00·16-s + 0.100·17-s − 0.333·18-s + 1.24·19-s − 0.727·22-s − 0.294·23-s − 0.577·24-s + 1.33·25-s + 1.73·26-s − 0.192·27-s − 1.12·29-s − 0.881·30-s + 0.538·31-s − 0.420·33-s − 0.100·34-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  =  1
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 1.41T + 2T^{2} \)
5 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
13 \( 1 + 6.24T + 13T^{2} \)
17 \( 1 - 0.414T + 17T^{2} \)
19 \( 1 - 5.41T + 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 + 6.07T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 9.48T + 37T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 6.82T + 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 - 4.65T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + 7.65T + 79T^{2} \)
83 \( 1 + 1.07T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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

−7.81536128051432299388540779343, −7.09605216694609755127372990738, −6.84727915170172598621879760082, −5.32623041096835230558497133466, −4.90542458372635834329249302918, −4.03332972064782946531072227623, −3.41170808101923195852329295083, −2.00559952499735569126155279467, −0.819393996952786040339090110820, 0, 0.819393996952786040339090110820, 2.00559952499735569126155279467, 3.41170808101923195852329295083, 4.03332972064782946531072227623, 4.90542458372635834329249302918, 5.32623041096835230558497133466, 6.84727915170172598621879760082, 7.09605216694609755127372990738, 7.81536128051432299388540779343

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