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  + 2.83·3-s − 2.22·5-s + 5.01·9-s − 1.51·11-s + 6.85·13-s − 6.28·15-s + 1.59·17-s − 2.36·19-s + 4.25·23-s − 0.0656·25-s + 5.70·27-s + 1.45·29-s + 2.09·31-s − 4.29·33-s + 7.96·37-s + 19.3·39-s − 41-s + 3.59·43-s − 11.1·45-s − 6.92·47-s + 4.51·51-s − 7.14·53-s + 3.36·55-s − 6.68·57-s − 3.14·59-s + 3.64·61-s − 15.2·65-s + ⋯
L(s)  = 1  + 1.63·3-s − 0.993·5-s + 1.67·9-s − 0.456·11-s + 1.89·13-s − 1.62·15-s + 0.386·17-s − 0.542·19-s + 0.887·23-s − 0.0131·25-s + 1.09·27-s + 0.269·29-s + 0.375·31-s − 0.746·33-s + 1.30·37-s + 3.10·39-s − 0.156·41-s + 0.547·43-s − 1.66·45-s − 1.01·47-s + 0.632·51-s − 0.982·53-s + 0.453·55-s − 0.886·57-s − 0.409·59-s + 0.466·61-s − 1.88·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.562821008$
$L(\frac12)$  $\approx$  $3.562821008$
$L(\frac{3}{2})$   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 \( 1 - 2.83T + 3T^{2} \)
5 \( 1 + 2.22T + 5T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 - 6.85T + 13T^{2} \)
17 \( 1 - 1.59T + 17T^{2} \)
19 \( 1 + 2.36T + 19T^{2} \)
23 \( 1 - 4.25T + 23T^{2} \)
29 \( 1 - 1.45T + 29T^{2} \)
31 \( 1 - 2.09T + 31T^{2} \)
37 \( 1 - 7.96T + 37T^{2} \)
43 \( 1 - 3.59T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 7.14T + 53T^{2} \)
59 \( 1 + 3.14T + 59T^{2} \)
61 \( 1 - 3.64T + 61T^{2} \)
67 \( 1 - 8.76T + 67T^{2} \)
71 \( 1 + 0.569T + 71T^{2} \)
73 \( 1 - 3.79T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 5.82T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−8.061242845401808110411936825255, −7.44968764560712790085684324288, −6.62110625845797429480657681378, −5.86408143316187524275388353037, −4.68467388279131935178503105122, −4.04826620248799991958500981102, −3.40525829846169667473339797456, −2.93786370252781191114343014132, −1.89945334566780144392103800330, −0.891164744877721232594817443471, 0.891164744877721232594817443471, 1.89945334566780144392103800330, 2.93786370252781191114343014132, 3.40525829846169667473339797456, 4.04826620248799991958500981102, 4.68467388279131935178503105122, 5.86408143316187524275388353037, 6.62110625845797429480657681378, 7.44968764560712790085684324288, 8.061242845401808110411936825255

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