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  − 0.515·3-s + 2.19·5-s − 2.73·9-s + 4.22·11-s + 3.63·13-s − 1.13·15-s − 7.98·17-s + 1.14·19-s − 7.43·23-s − 0.161·25-s + 2.95·27-s + 6.08·29-s − 1.33·31-s − 2.17·33-s − 2.87·37-s − 1.87·39-s − 41-s + 5.14·43-s − 6.01·45-s + 8.12·47-s + 4.11·51-s − 1.63·53-s + 9.28·55-s − 0.587·57-s − 2.97·59-s + 10.2·61-s + 8.00·65-s + ⋯
L(s)  = 1  − 0.297·3-s + 0.983·5-s − 0.911·9-s + 1.27·11-s + 1.00·13-s − 0.292·15-s − 1.93·17-s + 0.261·19-s − 1.55·23-s − 0.0323·25-s + 0.568·27-s + 1.12·29-s − 0.239·31-s − 0.378·33-s − 0.473·37-s − 0.300·39-s − 0.156·41-s + 0.784·43-s − 0.896·45-s + 1.18·47-s + 0.576·51-s − 0.224·53-s + 1.25·55-s − 0.0777·57-s − 0.387·59-s + 1.31·61-s + 0.992·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.104337354$
$L(\frac12)$  $\approx$  $2.104337354$
$L(\frac{3}{2})$   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 \( 1 + 0.515T + 3T^{2} \)
5 \( 1 - 2.19T + 5T^{2} \)
11 \( 1 - 4.22T + 11T^{2} \)
13 \( 1 - 3.63T + 13T^{2} \)
17 \( 1 + 7.98T + 17T^{2} \)
19 \( 1 - 1.14T + 19T^{2} \)
23 \( 1 + 7.43T + 23T^{2} \)
29 \( 1 - 6.08T + 29T^{2} \)
31 \( 1 + 1.33T + 31T^{2} \)
37 \( 1 + 2.87T + 37T^{2} \)
43 \( 1 - 5.14T + 43T^{2} \)
47 \( 1 - 8.12T + 47T^{2} \)
53 \( 1 + 1.63T + 53T^{2} \)
59 \( 1 + 2.97T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 2.46T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 9.06T + 79T^{2} \)
83 \( 1 - 7.51T + 83T^{2} \)
89 \( 1 + 17.9T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−7.975122720028078222677779118023, −6.74442308728575694725104691013, −6.40718617017307275817702938344, −5.92913866881740980560924495202, −5.18290986678522187825240996845, −4.21900203232766503407233534210, −3.64246626454295351779814827612, −2.44399044242890942171285233338, −1.87180961663245823767535219679, −0.72695519637306173515390691407, 0.72695519637306173515390691407, 1.87180961663245823767535219679, 2.44399044242890942171285233338, 3.64246626454295351779814827612, 4.21900203232766503407233534210, 5.18290986678522187825240996845, 5.92913866881740980560924495202, 6.40718617017307275817702938344, 6.74442308728575694725104691013, 7.975122720028078222677779118023

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