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  − 1.93·3-s − 0.475·5-s + 0.733·9-s + 0.608·11-s + 0.258·13-s + 0.918·15-s + 6.67·17-s − 8.55·19-s − 0.540·23-s − 4.77·25-s + 4.37·27-s + 9.17·29-s − 3.19·31-s − 1.17·33-s + 2.11·37-s − 0.499·39-s − 41-s − 9.45·43-s − 0.348·45-s − 5.21·47-s − 12.8·51-s + 11.1·53-s − 0.289·55-s + 16.5·57-s − 6.80·59-s + 1.40·61-s − 0.122·65-s + ⋯
L(s)  = 1  − 1.11·3-s − 0.212·5-s + 0.244·9-s + 0.183·11-s + 0.0716·13-s + 0.237·15-s + 1.61·17-s − 1.96·19-s − 0.112·23-s − 0.954·25-s + 0.842·27-s + 1.70·29-s − 0.573·31-s − 0.204·33-s + 0.347·37-s − 0.0799·39-s − 0.156·41-s − 1.44·43-s − 0.0519·45-s − 0.761·47-s − 1.80·51-s + 1.52·53-s − 0.0390·55-s + 2.19·57-s − 0.885·59-s + 0.179·61-s − 0.0152·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8906753432$
$L(\frac12)$  $\approx$  $0.8906753432$
$L(\frac{3}{2})$   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 \( 1 + 1.93T + 3T^{2} \)
5 \( 1 + 0.475T + 5T^{2} \)
11 \( 1 - 0.608T + 11T^{2} \)
13 \( 1 - 0.258T + 13T^{2} \)
17 \( 1 - 6.67T + 17T^{2} \)
19 \( 1 + 8.55T + 19T^{2} \)
23 \( 1 + 0.540T + 23T^{2} \)
29 \( 1 - 9.17T + 29T^{2} \)
31 \( 1 + 3.19T + 31T^{2} \)
37 \( 1 - 2.11T + 37T^{2} \)
43 \( 1 + 9.45T + 43T^{2} \)
47 \( 1 + 5.21T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 6.80T + 59T^{2} \)
61 \( 1 - 1.40T + 61T^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 - 3.72T + 71T^{2} \)
73 \( 1 + 9.31T + 73T^{2} \)
79 \( 1 - 5.46T + 79T^{2} \)
83 \( 1 - 0.971T + 83T^{2} \)
89 \( 1 - 9.78T + 89T^{2} \)
97 \( 1 - 3.26T + 97T^{2} \)
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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

−8.065823607289193033474160015993, −6.79281299479396906702135884830, −6.51742639049697437273572229902, −5.70952027301474829588086429880, −5.19184164905751295919824274981, −4.36512986367360194852469715004, −3.66304578859458639446080709341, −2.67488682750864764476171211210, −1.56901404098607660180193880952, −0.50647880441920067437649780952, 0.50647880441920067437649780952, 1.56901404098607660180193880952, 2.67488682750864764476171211210, 3.66304578859458639446080709341, 4.36512986367360194852469715004, 5.19184164905751295919824274981, 5.70952027301474829588086429880, 6.51742639049697437273572229902, 6.79281299479396906702135884830, 8.065823607289193033474160015993

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