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

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.72·3-s − 0.138·5-s − 0.00725·9-s + 2.69·11-s − 5.05·13-s + 0.239·15-s − 4.38·17-s + 5.73·19-s + 8.09·23-s − 4.98·25-s + 5.20·27-s + 3.49·29-s − 2.84·31-s − 4.65·33-s − 4.04·37-s + 8.73·39-s − 41-s − 10.8·43-s + 0.00100·45-s + 0.549·47-s + 7.58·51-s + 11.8·53-s − 0.372·55-s − 9.91·57-s − 9.85·59-s − 7.09·61-s + 0.698·65-s + ⋯
L(s)  = 1  − 0.998·3-s − 0.0618·5-s − 0.00241·9-s + 0.812·11-s − 1.40·13-s + 0.0617·15-s − 1.06·17-s + 1.31·19-s + 1.68·23-s − 0.996·25-s + 1.00·27-s + 0.648·29-s − 0.511·31-s − 0.811·33-s − 0.665·37-s + 1.39·39-s − 0.156·41-s − 1.65·43-s + 0.000149·45-s + 0.0801·47-s + 1.06·51-s + 1.63·53-s − 0.0502·55-s − 1.31·57-s − 1.28·59-s − 0.908·61-s + 0.0866·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9178708383$
$L(\frac12)$  $\approx$  $0.9178708383$
$L(\frac{3}{2})$   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 \( 1 + 1.72T + 3T^{2} \)
5 \( 1 + 0.138T + 5T^{2} \)
11 \( 1 - 2.69T + 11T^{2} \)
13 \( 1 + 5.05T + 13T^{2} \)
17 \( 1 + 4.38T + 17T^{2} \)
19 \( 1 - 5.73T + 19T^{2} \)
23 \( 1 - 8.09T + 23T^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
31 \( 1 + 2.84T + 31T^{2} \)
37 \( 1 + 4.04T + 37T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 0.549T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 9.85T + 59T^{2} \)
61 \( 1 + 7.09T + 61T^{2} \)
67 \( 1 + 2.80T + 67T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + 7.81T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 2.44T + 83T^{2} \)
89 \( 1 - 1.78T + 89T^{2} \)
97 \( 1 + 5.82T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.56035655655481037811591877879, −7.04863812010328189140208052806, −6.50763342677050451667717008194, −5.68755383444228591780348303860, −5.00221053416865192067725636429, −4.61810550261722777266751304585, −3.49572056903785092452335325532, −2.71244995863287951321867314389, −1.61208723723443628143101840840, −0.50388665197562363794364544963, 0.50388665197562363794364544963, 1.61208723723443628143101840840, 2.71244995863287951321867314389, 3.49572056903785092452335325532, 4.61810550261722777266751304585, 5.00221053416865192067725636429, 5.68755383444228591780348303860, 6.50763342677050451667717008194, 7.04863812010328189140208052806, 7.56035655655481037811591877879

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