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  − 1.40·3-s + 3.61·5-s − 1.02·9-s − 5.64·11-s + 6.54·13-s − 5.07·15-s + 3.39·17-s + 5.19·19-s − 2.81·23-s + 8.07·25-s + 5.65·27-s + 2.17·29-s + 6.26·31-s + 7.93·33-s + 0.641·37-s − 9.18·39-s − 41-s − 4.44·43-s − 3.71·45-s + 6.69·47-s − 4.76·51-s + 4.95·53-s − 20.4·55-s − 7.29·57-s − 11.0·59-s − 12.0·61-s + 23.6·65-s + ⋯
L(s)  = 1  − 0.810·3-s + 1.61·5-s − 0.342·9-s − 1.70·11-s + 1.81·13-s − 1.31·15-s + 0.822·17-s + 1.19·19-s − 0.585·23-s + 1.61·25-s + 1.08·27-s + 0.404·29-s + 1.12·31-s + 1.38·33-s + 0.105·37-s − 1.47·39-s − 0.156·41-s − 0.677·43-s − 0.554·45-s + 0.976·47-s − 0.666·51-s + 0.680·53-s − 2.75·55-s − 0.966·57-s − 1.43·59-s − 1.54·61-s + 2.93·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.139210754$
$L(\frac12)$  $\approx$  $2.139210754$
$L(\frac{3}{2})$   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 \( 1 + 1.40T + 3T^{2} \)
5 \( 1 - 3.61T + 5T^{2} \)
11 \( 1 + 5.64T + 11T^{2} \)
13 \( 1 - 6.54T + 13T^{2} \)
17 \( 1 - 3.39T + 17T^{2} \)
19 \( 1 - 5.19T + 19T^{2} \)
23 \( 1 + 2.81T + 23T^{2} \)
29 \( 1 - 2.17T + 29T^{2} \)
31 \( 1 - 6.26T + 31T^{2} \)
37 \( 1 - 0.641T + 37T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
47 \( 1 - 6.69T + 47T^{2} \)
53 \( 1 - 4.95T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + 8.70T + 71T^{2} \)
73 \( 1 - 0.422T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 - 5.06T + 83T^{2} \)
89 \( 1 + 1.56T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−7.944070383571078156748406405420, −6.94106820446540225465111997906, −6.03751050374712395167890023791, −5.84114106974453062059595334755, −5.34924068210010641464768801793, −4.60848407723921746480185649647, −3.23465588350775488703715475542, −2.73302996545315855080368529332, −1.62849833859940972570829587107, −0.790362379969394607797797583562, 0.790362379969394607797797583562, 1.62849833859940972570829587107, 2.73302996545315855080368529332, 3.23465588350775488703715475542, 4.60848407723921746480185649647, 5.34924068210010641464768801793, 5.84114106974453062059595334755, 6.03751050374712395167890023791, 6.94106820446540225465111997906, 7.944070383571078156748406405420

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