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  + 3.29·3-s + 3.10·5-s + 7.84·9-s + 3.57·11-s − 2.94·13-s + 10.2·15-s + 0.836·17-s + 1.07·19-s − 4.90·23-s + 4.62·25-s + 15.9·27-s − 3.82·29-s + 5.16·31-s + 11.7·33-s − 6.09·37-s − 9.68·39-s − 41-s + 7.59·43-s + 24.3·45-s − 13.2·47-s + 2.75·51-s − 3.67·53-s + 11.0·55-s + 3.54·57-s + 8.05·59-s + 6.21·61-s − 9.12·65-s + ⋯
L(s)  = 1  + 1.90·3-s + 1.38·5-s + 2.61·9-s + 1.07·11-s − 0.815·13-s + 2.63·15-s + 0.202·17-s + 0.246·19-s − 1.02·23-s + 0.924·25-s + 3.07·27-s − 0.709·29-s + 0.927·31-s + 2.05·33-s − 1.00·37-s − 1.55·39-s − 0.156·41-s + 1.15·43-s + 3.62·45-s − 1.93·47-s + 0.385·51-s − 0.504·53-s + 1.49·55-s + 0.468·57-s + 1.04·59-s + 0.796·61-s − 1.13·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.335472966$
$L(\frac12)$  $\approx$  $6.335472966$
$L(\frac{3}{2})$   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 \( 1 - 3.29T + 3T^{2} \)
5 \( 1 - 3.10T + 5T^{2} \)
11 \( 1 - 3.57T + 11T^{2} \)
13 \( 1 + 2.94T + 13T^{2} \)
17 \( 1 - 0.836T + 17T^{2} \)
19 \( 1 - 1.07T + 19T^{2} \)
23 \( 1 + 4.90T + 23T^{2} \)
29 \( 1 + 3.82T + 29T^{2} \)
31 \( 1 - 5.16T + 31T^{2} \)
37 \( 1 + 6.09T + 37T^{2} \)
43 \( 1 - 7.59T + 43T^{2} \)
47 \( 1 + 13.2T + 47T^{2} \)
53 \( 1 + 3.67T + 53T^{2} \)
59 \( 1 - 8.05T + 59T^{2} \)
61 \( 1 - 6.21T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 8.78T + 73T^{2} \)
79 \( 1 + 4.46T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 - 1.02T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−8.174862406537746063716526883736, −7.05825685601163559488331389696, −6.75092391492652711452811838371, −5.79351678035557013181568321133, −4.92393032276177498216879102240, −4.04529736826693046331793954508, −3.41435900614222095851375923259, −2.50177698677605173476335927867, −1.99874885146491878591735587718, −1.28169501407571667202204788676, 1.28169501407571667202204788676, 1.99874885146491878591735587718, 2.50177698677605173476335927867, 3.41435900614222095851375923259, 4.04529736826693046331793954508, 4.92393032276177498216879102240, 5.79351678035557013181568321133, 6.75092391492652711452811838371, 7.05825685601163559488331389696, 8.174862406537746063716526883736

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