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  − 1.73·3-s − 3.41·5-s + 0.00658·9-s + 1.67·11-s + 6.13·13-s + 5.91·15-s + 3.78·17-s + 3.86·19-s − 5.49·23-s + 6.65·25-s + 5.19·27-s − 0.172·29-s + 7.15·31-s − 2.90·33-s − 4.80·37-s − 10.6·39-s + 41-s + 0.845·43-s − 0.0224·45-s + 4.65·47-s − 6.57·51-s + 4.46·53-s − 5.72·55-s − 6.70·57-s − 3.38·59-s + 0.643·61-s − 20.9·65-s + ⋯
L(s)  = 1  − 1.00·3-s − 1.52·5-s + 0.00219·9-s + 0.505·11-s + 1.70·13-s + 1.52·15-s + 0.919·17-s + 0.887·19-s − 1.14·23-s + 1.33·25-s + 0.998·27-s − 0.0320·29-s + 1.28·31-s − 0.506·33-s − 0.789·37-s − 1.70·39-s + 0.156·41-s + 0.128·43-s − 0.00335·45-s + 0.679·47-s − 0.920·51-s + 0.613·53-s − 0.771·55-s − 0.888·57-s − 0.440·59-s + 0.0823·61-s − 2.59·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.029664069$
$L(\frac12)$  $\approx$  $1.029664069$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 + 1.73T + 3T^{2} \)
5 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 - 6.13T + 13T^{2} \)
17 \( 1 - 3.78T + 17T^{2} \)
19 \( 1 - 3.86T + 19T^{2} \)
23 \( 1 + 5.49T + 23T^{2} \)
29 \( 1 + 0.172T + 29T^{2} \)
31 \( 1 - 7.15T + 31T^{2} \)
37 \( 1 + 4.80T + 37T^{2} \)
43 \( 1 - 0.845T + 43T^{2} \)
47 \( 1 - 4.65T + 47T^{2} \)
53 \( 1 - 4.46T + 53T^{2} \)
59 \( 1 + 3.38T + 59T^{2} \)
61 \( 1 - 0.643T + 61T^{2} \)
67 \( 1 + 2.25T + 67T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 - 9.42T + 73T^{2} \)
79 \( 1 - 3.64T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 6.83T + 89T^{2} \)
97 \( 1 + 0.615T + 97T^{2} \)
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

−7.87486222479285391862831329598, −7.12136439906487729583385741354, −6.34406327517353369235537934635, −5.83506618176386668611071429432, −5.08707192654236210624533885240, −4.15773097452059520824270756374, −3.70539659146532786099615287350, −2.95794458095094828695317582686, −1.32570060818648344180165015796, −0.60292247404132829916732644767, 0.60292247404132829916732644767, 1.32570060818648344180165015796, 2.95794458095094828695317582686, 3.70539659146532786099615287350, 4.15773097452059520824270756374, 5.08707192654236210624533885240, 5.83506618176386668611071429432, 6.34406327517353369235537934635, 7.12136439906487729583385741354, 7.87486222479285391862831329598

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