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  + 2.70·3-s + 0.469·5-s + 4.32·9-s + 4.72·11-s + 1.70·13-s + 1.27·15-s − 3.62·17-s + 0.482·19-s − 0.0771·23-s − 4.77·25-s + 3.58·27-s + 0.891·29-s + 9.17·31-s + 12.7·33-s + 2.10·37-s + 4.60·39-s + 41-s + 6.61·43-s + 2.03·45-s + 4.76·47-s − 9.80·51-s − 12.1·53-s + 2.21·55-s + 1.30·57-s + 8.46·59-s + 1.21·61-s + 0.798·65-s + ⋯
L(s)  = 1  + 1.56·3-s + 0.209·5-s + 1.44·9-s + 1.42·11-s + 0.472·13-s + 0.327·15-s − 0.878·17-s + 0.110·19-s − 0.0160·23-s − 0.955·25-s + 0.690·27-s + 0.165·29-s + 1.64·31-s + 2.22·33-s + 0.345·37-s + 0.737·39-s + 0.156·41-s + 1.00·43-s + 0.302·45-s + 0.695·47-s − 1.37·51-s − 1.66·53-s + 0.299·55-s + 0.173·57-s + 1.10·59-s + 0.155·61-s + 0.0990·65-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  =  $(2,\ 8036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.579799756$
$L(\frac12)$  $\approx$  $4.579799756$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 - 2.70T + 3T^{2} \)
5 \( 1 - 0.469T + 5T^{2} \)
11 \( 1 - 4.72T + 11T^{2} \)
13 \( 1 - 1.70T + 13T^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
19 \( 1 - 0.482T + 19T^{2} \)
23 \( 1 + 0.0771T + 23T^{2} \)
29 \( 1 - 0.891T + 29T^{2} \)
31 \( 1 - 9.17T + 31T^{2} \)
37 \( 1 - 2.10T + 37T^{2} \)
43 \( 1 - 6.61T + 43T^{2} \)
47 \( 1 - 4.76T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 8.46T + 59T^{2} \)
61 \( 1 - 1.21T + 61T^{2} \)
67 \( 1 + 3.96T + 67T^{2} \)
71 \( 1 - 5.12T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 1.02T + 83T^{2} \)
89 \( 1 + 3.74T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−7.926164493371797563143233624629, −7.31225098155651705404263012570, −6.44650404583422205718773964562, −6.03976054072190731316381168862, −4.71802841625566723242219700852, −4.07779396810916452678271684221, −3.52498928293965356242351515912, −2.63739401406014886625076782712, −1.95351335419280275535001109835, −1.04276472495611986606309500530, 1.04276472495611986606309500530, 1.95351335419280275535001109835, 2.63739401406014886625076782712, 3.52498928293965356242351515912, 4.07779396810916452678271684221, 4.71802841625566723242219700852, 6.03976054072190731316381168862, 6.44650404583422205718773964562, 7.31225098155651705404263012570, 7.926164493371797563143233624629

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