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  + 2.47·3-s − 3.49·5-s + 3.13·9-s − 0.886·11-s − 6.85·13-s − 8.66·15-s + 3.02·17-s + 6.07·19-s − 4.52·23-s + 7.23·25-s + 0.340·27-s − 7.22·29-s + 9.72·31-s − 2.19·33-s + 7.05·37-s − 16.9·39-s − 41-s + 4.25·43-s − 10.9·45-s − 11.7·47-s + 7.50·51-s + 6.23·53-s + 3.10·55-s + 15.0·57-s + 0.453·59-s − 1.80·61-s + 23.9·65-s + ⋯
L(s)  = 1  + 1.43·3-s − 1.56·5-s + 1.04·9-s − 0.267·11-s − 1.90·13-s − 2.23·15-s + 0.734·17-s + 1.39·19-s − 0.943·23-s + 1.44·25-s + 0.0654·27-s − 1.34·29-s + 1.74·31-s − 0.382·33-s + 1.16·37-s − 2.71·39-s − 0.156·41-s + 0.649·43-s − 1.63·45-s − 1.71·47-s + 1.05·51-s + 0.856·53-s + 0.418·55-s + 1.99·57-s + 0.0590·59-s − 0.231·61-s + 2.97·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.954780416$
$L(\frac12)$  $\approx$  $1.954780416$
$L(\frac{3}{2})$   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 \( 1 - 2.47T + 3T^{2} \)
5 \( 1 + 3.49T + 5T^{2} \)
11 \( 1 + 0.886T + 11T^{2} \)
13 \( 1 + 6.85T + 13T^{2} \)
17 \( 1 - 3.02T + 17T^{2} \)
19 \( 1 - 6.07T + 19T^{2} \)
23 \( 1 + 4.52T + 23T^{2} \)
29 \( 1 + 7.22T + 29T^{2} \)
31 \( 1 - 9.72T + 31T^{2} \)
37 \( 1 - 7.05T + 37T^{2} \)
43 \( 1 - 4.25T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 6.23T + 53T^{2} \)
59 \( 1 - 0.453T + 59T^{2} \)
61 \( 1 + 1.80T + 61T^{2} \)
67 \( 1 - 0.386T + 67T^{2} \)
71 \( 1 - 0.406T + 71T^{2} \)
73 \( 1 - 2.47T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 7.79T + 83T^{2} \)
89 \( 1 + 5.14T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−7.928910010072881820298936326142, −7.52084242668662150035762427471, −6.87145573551640012318639778456, −5.60974371109891341310155818119, −4.76547053867026184199840822996, −4.14569688486634342279626142909, −3.34643173780320107810804756610, −2.89483391851068203108329180172, −2.05089984535897565320157195521, −0.61908719078495041650121021349, 0.61908719078495041650121021349, 2.05089984535897565320157195521, 2.89483391851068203108329180172, 3.34643173780320107810804756610, 4.14569688486634342279626142909, 4.76547053867026184199840822996, 5.60974371109891341310155818119, 6.87145573551640012318639778456, 7.52084242668662150035762427471, 7.928910010072881820298936326142

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