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  − 2.32·3-s + 1.16·5-s + 2.40·9-s + 0.577·11-s + 2.18·13-s − 2.70·15-s + 6.65·17-s − 2.12·19-s + 8.32·23-s − 3.64·25-s + 1.37·27-s − 6.30·29-s − 1.38·31-s − 1.34·33-s + 2.86·37-s − 5.07·39-s − 41-s + 6.21·43-s + 2.80·45-s + 12.3·47-s − 15.4·51-s + 13.0·53-s + 0.672·55-s + 4.94·57-s − 3.27·59-s + 5.15·61-s + 2.54·65-s + ⋯
L(s)  = 1  − 1.34·3-s + 0.520·5-s + 0.802·9-s + 0.174·11-s + 0.605·13-s − 0.699·15-s + 1.61·17-s − 0.488·19-s + 1.73·23-s − 0.728·25-s + 0.265·27-s − 1.16·29-s − 0.249·31-s − 0.233·33-s + 0.471·37-s − 0.812·39-s − 0.156·41-s + 0.947·43-s + 0.417·45-s + 1.80·47-s − 2.16·51-s + 1.79·53-s + 0.0907·55-s + 0.655·57-s − 0.426·59-s + 0.660·61-s + 0.315·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.514721090$
$L(\frac12)$  $\approx$  $1.514721090$
$L(\frac{3}{2})$   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 \( 1 + 2.32T + 3T^{2} \)
5 \( 1 - 1.16T + 5T^{2} \)
11 \( 1 - 0.577T + 11T^{2} \)
13 \( 1 - 2.18T + 13T^{2} \)
17 \( 1 - 6.65T + 17T^{2} \)
19 \( 1 + 2.12T + 19T^{2} \)
23 \( 1 - 8.32T + 23T^{2} \)
29 \( 1 + 6.30T + 29T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
43 \( 1 - 6.21T + 43T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 + 3.27T + 59T^{2} \)
61 \( 1 - 5.15T + 61T^{2} \)
67 \( 1 + 5.35T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 - 8.30T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 - 6.56T + 89T^{2} \)
97 \( 1 + 2.09T + 97T^{2} \)
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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

−7.50835494455301593202136294049, −7.15229355431700799795713001109, −6.16621071541714007585157606947, −5.71298526820566962561823972423, −5.38376880642922259903050290018, −4.41350554087611557071075395538, −3.63269741768165680032451159189, −2.63915971290070752399756380668, −1.45170824075511595252636658981, −0.71623825345992442071301437704, 0.71623825345992442071301437704, 1.45170824075511595252636658981, 2.63915971290070752399756380668, 3.63269741768165680032451159189, 4.41350554087611557071075395538, 5.38376880642922259903050290018, 5.71298526820566962561823972423, 6.16621071541714007585157606947, 7.15229355431700799795713001109, 7.50835494455301593202136294049

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