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  + 2.36·3-s + 2.42·5-s + 2.60·9-s − 5.39·11-s + 6.20·13-s + 5.74·15-s − 2.01·17-s + 6.62·19-s + 1.84·23-s + 0.877·25-s − 0.933·27-s + 6.47·29-s − 8.80·31-s − 12.7·33-s + 3.09·37-s + 14.6·39-s − 41-s + 10.5·43-s + 6.31·45-s + 3.78·47-s − 4.77·51-s − 1.87·53-s − 13.0·55-s + 15.6·57-s + 3.79·59-s + 6.72·61-s + 15.0·65-s + ⋯
L(s)  = 1  + 1.36·3-s + 1.08·5-s + 0.868·9-s − 1.62·11-s + 1.71·13-s + 1.48·15-s − 0.488·17-s + 1.52·19-s + 0.385·23-s + 0.175·25-s − 0.179·27-s + 1.20·29-s − 1.58·31-s − 2.22·33-s + 0.508·37-s + 2.35·39-s − 0.156·41-s + 1.61·43-s + 0.941·45-s + 0.552·47-s − 0.668·51-s − 0.257·53-s − 1.76·55-s + 2.07·57-s + 0.494·59-s + 0.861·61-s + 1.86·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.501343580$
$L(\frac12)$  $\approx$  $4.501343580$
$L(\frac{3}{2})$   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 \( 1 - 2.36T + 3T^{2} \)
5 \( 1 - 2.42T + 5T^{2} \)
11 \( 1 + 5.39T + 11T^{2} \)
13 \( 1 - 6.20T + 13T^{2} \)
17 \( 1 + 2.01T + 17T^{2} \)
19 \( 1 - 6.62T + 19T^{2} \)
23 \( 1 - 1.84T + 23T^{2} \)
29 \( 1 - 6.47T + 29T^{2} \)
31 \( 1 + 8.80T + 31T^{2} \)
37 \( 1 - 3.09T + 37T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 3.78T + 47T^{2} \)
53 \( 1 + 1.87T + 53T^{2} \)
59 \( 1 - 3.79T + 59T^{2} \)
61 \( 1 - 6.72T + 61T^{2} \)
67 \( 1 + 3.50T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 1.71T + 73T^{2} \)
79 \( 1 + 0.277T + 79T^{2} \)
83 \( 1 + 6.38T + 83T^{2} \)
89 \( 1 + 1.19T + 89T^{2} \)
97 \( 1 + 9.32T + 97T^{2} \)
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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

−7.84663536865244553763916871192, −7.42175194873781820619111066045, −6.42897339421765789728057666162, −5.64931226258010244691283327395, −5.22576357754608495648956002370, −4.06486912435668663164027460280, −3.29124646302107825135274557209, −2.65467850992013031183798920096, −2.02488135188541447097495659075, −1.02014045120410202561911441466, 1.02014045120410202561911441466, 2.02488135188541447097495659075, 2.65467850992013031183798920096, 3.29124646302107825135274557209, 4.06486912435668663164027460280, 5.22576357754608495648956002370, 5.64931226258010244691283327395, 6.42897339421765789728057666162, 7.42175194873781820619111066045, 7.84663536865244553763916871192

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