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  + 0.449·3-s − 0.787·5-s − 2.79·9-s + 3.10·11-s + 5.59·13-s − 0.354·15-s + 5.38·17-s + 4.82·19-s + 7.02·23-s − 4.37·25-s − 2.60·27-s − 7.52·29-s + 4.90·31-s + 1.39·33-s − 9.21·37-s + 2.51·39-s − 41-s + 11.7·43-s + 2.20·45-s − 5.56·47-s + 2.41·51-s + 5.94·53-s − 2.44·55-s + 2.16·57-s − 10.2·59-s + 4.89·61-s − 4.40·65-s + ⋯
L(s)  = 1  + 0.259·3-s − 0.352·5-s − 0.932·9-s + 0.935·11-s + 1.55·13-s − 0.0914·15-s + 1.30·17-s + 1.10·19-s + 1.46·23-s − 0.875·25-s − 0.501·27-s − 1.39·29-s + 0.881·31-s + 0.242·33-s − 1.51·37-s + 0.402·39-s − 0.156·41-s + 1.79·43-s + 0.328·45-s − 0.811·47-s + 0.338·51-s + 0.816·53-s − 0.329·55-s + 0.287·57-s − 1.32·59-s + 0.626·61-s − 0.546·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.476287763$
$L(\frac12)$  $\approx$  $2.476287763$
$L(\frac{3}{2})$   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 \( 1 - 0.449T + 3T^{2} \)
5 \( 1 + 0.787T + 5T^{2} \)
11 \( 1 - 3.10T + 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 - 5.38T + 17T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 - 7.02T + 23T^{2} \)
29 \( 1 + 7.52T + 29T^{2} \)
31 \( 1 - 4.90T + 31T^{2} \)
37 \( 1 + 9.21T + 37T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + 5.56T + 47T^{2} \)
53 \( 1 - 5.94T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 4.89T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + 1.39T + 71T^{2} \)
73 \( 1 + 4.28T + 73T^{2} \)
79 \( 1 - 8.10T + 79T^{2} \)
83 \( 1 - 2.10T + 83T^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 - 2.23T + 97T^{2} \)
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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

−7.88086355404094842900474786009, −7.23705768032480040936752767542, −6.41991495483514319473899062703, −5.67909658758860547728984522043, −5.25244335015389600186036487094, −3.97928669407501849940603899834, −3.52259413794071495314366399307, −2.93233048248120963295018723195, −1.61076436083950247559149220916, −0.827211650455433768624104700014, 0.827211650455433768624104700014, 1.61076436083950247559149220916, 2.93233048248120963295018723195, 3.52259413794071495314366399307, 3.97928669407501849940603899834, 5.25244335015389600186036487094, 5.67909658758860547728984522043, 6.41991495483514319473899062703, 7.23705768032480040936752767542, 7.88086355404094842900474786009

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