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  + 1.53·3-s + 1.74·5-s − 0.644·9-s + 1.47·11-s − 0.836·13-s + 2.68·15-s + 4.36·17-s − 0.873·19-s + 3.62·23-s − 1.93·25-s − 5.59·27-s + 5.63·29-s + 9.19·31-s + 2.25·33-s + 2.15·37-s − 1.28·39-s − 41-s − 1.50·43-s − 1.12·45-s + 8.53·47-s + 6.70·51-s + 7.12·53-s + 2.57·55-s − 1.34·57-s − 12.9·59-s − 12.0·61-s − 1.46·65-s + ⋯
L(s)  = 1  + 0.886·3-s + 0.782·5-s − 0.214·9-s + 0.443·11-s − 0.231·13-s + 0.693·15-s + 1.05·17-s − 0.200·19-s + 0.754·23-s − 0.387·25-s − 1.07·27-s + 1.04·29-s + 1.65·31-s + 0.392·33-s + 0.354·37-s − 0.205·39-s − 0.156·41-s − 0.228·43-s − 0.168·45-s + 1.24·47-s + 0.938·51-s + 0.978·53-s + 0.347·55-s − 0.177·57-s − 1.68·59-s − 1.54·61-s − 0.181·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.558341460$
$L(\frac12)$  $\approx$  $3.558341460$
$L(\frac{3}{2})$   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 \( 1 - 1.53T + 3T^{2} \)
5 \( 1 - 1.74T + 5T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + 0.836T + 13T^{2} \)
17 \( 1 - 4.36T + 17T^{2} \)
19 \( 1 + 0.873T + 19T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 - 5.63T + 29T^{2} \)
31 \( 1 - 9.19T + 31T^{2} \)
37 \( 1 - 2.15T + 37T^{2} \)
43 \( 1 + 1.50T + 43T^{2} \)
47 \( 1 - 8.53T + 47T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 + 4.82T + 71T^{2} \)
73 \( 1 + 0.0120T + 73T^{2} \)
79 \( 1 - 2.55T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 - 3.91T + 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−7.88840198402777764719333458251, −7.29248490798376662548230810601, −6.30710067903309636554121037659, −5.91305575885123204229300898278, −5.00913136603442480502962746118, −4.25249821794079477178549099420, −3.24226662254494942132070197421, −2.76282350094754118069462466556, −1.90554367713335633353684353092, −0.919613617761374012079029490437, 0.919613617761374012079029490437, 1.90554367713335633353684353092, 2.76282350094754118069462466556, 3.24226662254494942132070197421, 4.25249821794079477178549099420, 5.00913136603442480502962746118, 5.91305575885123204229300898278, 6.30710067903309636554121037659, 7.29248490798376662548230810601, 7.88840198402777764719333458251

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