Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7^{2} $
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-s − 3·3-s + 4-s + 5-s − 3·6-s + 8-s + 6·9-s + 10-s − 2·11-s − 3·12-s − 3·15-s + 16-s + 4·17-s + 6·18-s + 6·19-s + 20-s − 2·22-s + 3·23-s − 3·24-s + 25-s − 9·27-s + 9·29-s − 3·30-s + 4·31-s + 32-s + 6·33-s + 4·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s − 0.866·12-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 1.41·18-s + 1.37·19-s + 0.223·20-s − 0.426·22-s + 0.625·23-s − 0.612·24-s + 1/5·25-s − 1.73·27-s + 1.67·29-s − 0.547·30-s + 0.718·31-s + 0.176·32-s + 1.04·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{490} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 490,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.40933$
$L(\frac12)$  $\approx$  $1.40933$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−11.06289048806052475370595948438, −10.36383578936153834327307787127, −9.635128409981478362582371513154, −7.916325009885536263558334349206, −6.89394037901040901071472214534, −6.09157415694094013326272320913, −5.26789391708575690709750011159, −4.72473630829615747045956251199, −3.08854077694740486482752774267, −1.16479363083596500346798967246, 1.16479363083596500346798967246, 3.08854077694740486482752774267, 4.72473630829615747045956251199, 5.26789391708575690709750011159, 6.09157415694094013326272320913, 6.89394037901040901071472214534, 7.916325009885536263558334349206, 9.635128409981478362582371513154, 10.36383578936153834327307787127, 11.06289048806052475370595948438

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