Properties

Degree 2
Conductor $ 2^{2} \cdot 7^{2} \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 2·9-s + 3·11-s − 2·13-s + 15-s − 17-s + 3·19-s − 5·23-s − 4·25-s − 5·27-s − 2·29-s − 5·31-s + 3·33-s + 7·37-s − 2·39-s − 41-s + 4·43-s − 2·45-s + 3·47-s − 51-s − 3·53-s + 3·55-s + 3·57-s − 5·59-s − 3·61-s − 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.904·11-s − 0.554·13-s + 0.258·15-s − 0.242·17-s + 0.688·19-s − 1.04·23-s − 4/5·25-s − 0.962·27-s − 0.371·29-s − 0.898·31-s + 0.522·33-s + 1.15·37-s − 0.320·39-s − 0.156·41-s + 0.609·43-s − 0.298·45-s + 0.437·47-s − 0.140·51-s − 0.412·53-s + 0.404·55-s + 0.397·57-s − 0.650·59-s − 0.384·61-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  =  1
Selberg data  =  $(2,\ 8036,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−7.61943749813729177520105007801, −6.85289988751694553617350773896, −5.87337517143587168019418353841, −5.70438425232851408929291702679, −4.52450620716545246028760127701, −3.86446713775601186770675365542, −3.03381507596417230763587216299, −2.26083267815371982892389756080, −1.47388265660226401463321390245, 0, 1.47388265660226401463321390245, 2.26083267815371982892389756080, 3.03381507596417230763587216299, 3.86446713775601186770675365542, 4.52450620716545246028760127701, 5.70438425232851408929291702679, 5.87337517143587168019418353841, 6.85289988751694553617350773896, 7.61943749813729177520105007801

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