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  + 3-s + 5-s − 2·9-s − 5·11-s + 2·13-s + 15-s + 3·17-s + 3·19-s − 23-s − 4·25-s − 5·27-s − 10·29-s + 11·31-s − 5·33-s + 7·37-s + 2·39-s + 41-s + 8·43-s − 2·45-s + 7·47-s + 3·51-s − 11·53-s − 5·55-s + 3·57-s + 7·59-s + 61-s + 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.50·11-s + 0.554·13-s + 0.258·15-s + 0.727·17-s + 0.688·19-s − 0.208·23-s − 4/5·25-s − 0.962·27-s − 1.85·29-s + 1.97·31-s − 0.870·33-s + 1.15·37-s + 0.320·39-s + 0.156·41-s + 1.21·43-s − 0.298·45-s + 1.02·47-s + 0.420·51-s − 1.51·53-s − 0.674·55-s + 0.397·57-s + 0.911·59-s + 0.128·61-s + 0.248·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.283022571$
$L(\frac12)$  $\approx$  $2.283022571$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.77593179438390054962450948682, −7.53016255729802160081509384691, −6.22827256931015434938578234610, −5.74240001366221175143832145237, −5.23035060020067689268665763590, −4.19799837391576246663386741899, −3.32322767400617550083781111066, −2.67916537566897855360992727818, −2.00412240876726749654446199912, −0.70520324392681296805677422701, 0.70520324392681296805677422701, 2.00412240876726749654446199912, 2.67916537566897855360992727818, 3.32322767400617550083781111066, 4.19799837391576246663386741899, 5.23035060020067689268665763590, 5.74240001366221175143832145237, 6.22827256931015434938578234610, 7.53016255729802160081509384691, 7.77593179438390054962450948682

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