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  + 3.09·3-s + 2.53·5-s + 6.56·9-s + 2.41·11-s + 0.891·13-s + 7.83·15-s − 1.35·17-s + 7.44·19-s + 7.55·23-s + 1.42·25-s + 11.0·27-s − 9.21·29-s − 7.97·31-s + 7.46·33-s − 3.88·37-s + 2.75·39-s − 41-s − 10.1·43-s + 16.6·45-s + 11.3·47-s − 4.17·51-s + 6.76·53-s + 6.11·55-s + 23.0·57-s + 5.15·59-s − 12.0·61-s + 2.26·65-s + ⋯
L(s)  = 1  + 1.78·3-s + 1.13·5-s + 2.18·9-s + 0.727·11-s + 0.247·13-s + 2.02·15-s − 0.327·17-s + 1.70·19-s + 1.57·23-s + 0.284·25-s + 2.12·27-s − 1.71·29-s − 1.43·31-s + 1.30·33-s − 0.638·37-s + 0.441·39-s − 0.156·41-s − 1.54·43-s + 2.48·45-s + 1.64·47-s − 0.585·51-s + 0.929·53-s + 0.825·55-s + 3.04·57-s + 0.671·59-s − 1.53·61-s + 0.280·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.028605865$
$L(\frac12)$  $\approx$  $6.028605865$
$L(\frac{3}{2})$   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 \( 1 - 3.09T + 3T^{2} \)
5 \( 1 - 2.53T + 5T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
13 \( 1 - 0.891T + 13T^{2} \)
17 \( 1 + 1.35T + 17T^{2} \)
19 \( 1 - 7.44T + 19T^{2} \)
23 \( 1 - 7.55T + 23T^{2} \)
29 \( 1 + 9.21T + 29T^{2} \)
31 \( 1 + 7.97T + 31T^{2} \)
37 \( 1 + 3.88T + 37T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 6.76T + 53T^{2} \)
59 \( 1 - 5.15T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 2.32T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 5.32T + 83T^{2} \)
89 \( 1 + 1.69T + 89T^{2} \)
97 \( 1 - 7.08T + 97T^{2} \)
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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

−7.78892309128426159218478223716, −7.23971060152866474484769195034, −6.73741206224597084278527834439, −5.63323929400022296569410950005, −5.10805285095015335046931553935, −3.91059457753930479860091170551, −3.44956182679336725209325895211, −2.67104699495101908944584803921, −1.83540436752258832047687192759, −1.28623113133546361679160960266, 1.28623113133546361679160960266, 1.83540436752258832047687192759, 2.67104699495101908944584803921, 3.44956182679336725209325895211, 3.91059457753930479860091170551, 5.10805285095015335046931553935, 5.63323929400022296569410950005, 6.73741206224597084278527834439, 7.23971060152866474484769195034, 7.78892309128426159218478223716

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