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  − 1.27·3-s + 4.37·5-s − 1.37·9-s + 6.01·11-s − 5.59·13-s − 5.57·15-s + 6.15·17-s + 5.17·19-s − 4.66·23-s + 14.1·25-s + 5.57·27-s − 1.77·29-s − 4.02·31-s − 7.66·33-s + 8.25·37-s + 7.13·39-s − 41-s + 0.831·43-s − 6.01·45-s + 2.72·47-s − 7.84·51-s − 0.906·53-s + 26.3·55-s − 6.59·57-s + 7.92·59-s − 5.68·61-s − 24.4·65-s + ⋯
L(s)  = 1  − 0.735·3-s + 1.95·5-s − 0.458·9-s + 1.81·11-s − 1.55·13-s − 1.43·15-s + 1.49·17-s + 1.18·19-s − 0.972·23-s + 2.82·25-s + 1.07·27-s − 0.330·29-s − 0.722·31-s − 1.33·33-s + 1.35·37-s + 1.14·39-s − 0.156·41-s + 0.126·43-s − 0.897·45-s + 0.396·47-s − 1.09·51-s − 0.124·53-s + 3.54·55-s − 0.873·57-s + 1.03·59-s − 0.728·61-s − 3.03·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.643225166$
$L(\frac12)$  $\approx$  $2.643225166$
$L(\frac{3}{2})$   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 \( 1 + 1.27T + 3T^{2} \)
5 \( 1 - 4.37T + 5T^{2} \)
11 \( 1 - 6.01T + 11T^{2} \)
13 \( 1 + 5.59T + 13T^{2} \)
17 \( 1 - 6.15T + 17T^{2} \)
19 \( 1 - 5.17T + 19T^{2} \)
23 \( 1 + 4.66T + 23T^{2} \)
29 \( 1 + 1.77T + 29T^{2} \)
31 \( 1 + 4.02T + 31T^{2} \)
37 \( 1 - 8.25T + 37T^{2} \)
43 \( 1 - 0.831T + 43T^{2} \)
47 \( 1 - 2.72T + 47T^{2} \)
53 \( 1 + 0.906T + 53T^{2} \)
59 \( 1 - 7.92T + 59T^{2} \)
61 \( 1 + 5.68T + 61T^{2} \)
67 \( 1 + 0.0872T + 67T^{2} \)
71 \( 1 + 8.10T + 71T^{2} \)
73 \( 1 + 5.60T + 73T^{2} \)
79 \( 1 - 0.401T + 79T^{2} \)
83 \( 1 + 7.91T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + 7.58T + 97T^{2} \)
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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

−7.60528820536882034886229786983, −7.00200934709458327708372249039, −6.16417994845550342857057169302, −5.81874219466190369224926407128, −5.30355412665253148513540379059, −4.55873326022564475678889993306, −3.38230736752382911096071384161, −2.55354465673741852941835719587, −1.65493564086321759409824520635, −0.892726985270966094150608044726, 0.892726985270966094150608044726, 1.65493564086321759409824520635, 2.55354465673741852941835719587, 3.38230736752382911096071384161, 4.55873326022564475678889993306, 5.30355412665253148513540379059, 5.81874219466190369224926407128, 6.16417994845550342857057169302, 7.00200934709458327708372249039, 7.60528820536882034886229786983

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