Properties

Degree $2$
Conductor $490$
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 − 2·3-s + 4-s − 5-s − 2·6-s + 8-s + 9-s − 10-s + 3·11-s − 2·12-s + 5·13-s + 2·15-s + 16-s + 6·17-s + 18-s − 19-s − 20-s + 3·22-s + 3·23-s − 2·24-s + 25-s + 5·26-s + 4·27-s − 6·29-s + 2·30-s − 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s − 0.577·12-s + 1.38·13-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.639·22-s + 0.625·23-s − 0.408·24-s + 1/5·25-s + 0.980·26-s + 0.769·27-s − 1.11·29-s + 0.365·30-s − 0.718·31-s + 0.176·32-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

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{490} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.521299208\)
\(L(\frac12)\) \(\approx\) \(1.521299208\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.81042978415589, −18.72805747076408, −18.25343762021672, −16.98551169655350, −16.69333387513004, −15.99600081230884, −14.95776425220329, −14.37909582229143, −13.27641870580694, −12.55122751025343, −11.65639467631427, −11.30027965538864, −10.49203076608856, −9.252183449285188, −8.089955271560401, −6.982458199053828, −6.071658603338646, −5.471843514426487, −4.241549060726637, −3.324600142303528, −1.194603962629732, 1.194603962629732, 3.324600142303528, 4.241549060726637, 5.471843514426487, 6.071658603338646, 6.982458199053828, 8.089955271560401, 9.252183449285188, 10.49203076608856, 11.30027965538864, 11.65639467631427, 12.55122751025343, 13.27641870580694, 14.37909582229143, 14.95776425220329, 15.99600081230884, 16.69333387513004, 16.98551169655350, 18.25343762021672, 18.72805747076408, 19.81042978415589

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