Properties

Degree 2
Conductor $ 2^{2} \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  − 2.15·3-s − 0.0591·5-s + 1.63·9-s − 5.19·11-s + 1.33·13-s + 0.127·15-s − 5.25·17-s − 6.42·19-s − 5.30·23-s − 4.99·25-s + 2.94·27-s − 2.05·29-s − 4.21·31-s + 11.1·33-s − 6.95·37-s − 2.87·39-s − 41-s − 4.70·43-s − 0.0964·45-s + 4.09·47-s + 11.3·51-s + 2.41·53-s + 0.307·55-s + 13.8·57-s − 4.59·59-s − 4.59·61-s − 0.0790·65-s + ⋯
L(s)  = 1  − 1.24·3-s − 0.0264·5-s + 0.543·9-s − 1.56·11-s + 0.370·13-s + 0.0328·15-s − 1.27·17-s − 1.47·19-s − 1.10·23-s − 0.999·25-s + 0.566·27-s − 0.382·29-s − 0.756·31-s + 1.94·33-s − 1.14·37-s − 0.460·39-s − 0.156·41-s − 0.717·43-s − 0.0143·45-s + 0.597·47-s + 1.58·51-s + 0.331·53-s + 0.0414·55-s + 1.83·57-s − 0.597·59-s − 0.588·61-s − 0.00980·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8036\)    =    \(2^{2} \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.03003730623$
$L(\frac12)$  $\approx$  $0.03003730623$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 + 2.15T + 3T^{2} \)
5 \( 1 + 0.0591T + 5T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 - 1.33T + 13T^{2} \)
17 \( 1 + 5.25T + 17T^{2} \)
19 \( 1 + 6.42T + 19T^{2} \)
23 \( 1 + 5.30T + 23T^{2} \)
29 \( 1 + 2.05T + 29T^{2} \)
31 \( 1 + 4.21T + 31T^{2} \)
37 \( 1 + 6.95T + 37T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 - 4.09T + 47T^{2} \)
53 \( 1 - 2.41T + 53T^{2} \)
59 \( 1 + 4.59T + 59T^{2} \)
61 \( 1 + 4.59T + 61T^{2} \)
67 \( 1 + 9.74T + 67T^{2} \)
71 \( 1 - 2.36T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 + 17.6T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 - 0.670T + 89T^{2} \)
97 \( 1 + 3.02T + 97T^{2} \)
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

−7.81208584952041719486307464410, −6.99001944614401343072787775180, −6.28765886324844848411561911589, −5.79686087448955201447541575583, −5.13701339868055428066635417031, −4.46339353605032648704172548976, −3.69476303573132851458635866451, −2.47777955305322559812610054087, −1.79917397684466856545841386334, −0.085823669419187712874194747850, 0.085823669419187712874194747850, 1.79917397684466856545841386334, 2.47777955305322559812610054087, 3.69476303573132851458635866451, 4.46339353605032648704172548976, 5.13701339868055428066635417031, 5.79686087448955201447541575583, 6.28765886324844848411561911589, 6.99001944614401343072787775180, 7.81208584952041719486307464410

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