Properties

Degree $2$
Conductor $490$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

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 − 4·11-s − 2·12-s − 2·13-s + 2·15-s + 16-s − 8·17-s + 18-s + 6·19-s − 20-s − 4·22-s − 4·23-s − 2·24-s + 25-s − 2·26-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 − 1.20·11-s − 0.577·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.852·22-s − 0.834·23-s − 0.408·24-s + 1/5·25-s − 0.392·26-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: \(1\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−19.40646700484665, −18.21374358987264, −17.85140258098576, −16.90557225798763, −15.99098091199238, −15.71634337737745, −14.73688343800399, −13.67165355575487, −12.98330226071648, −12.12691150722133, −11.45340338361601, −10.86614564966970, −9.987626462691324, −8.621807622472038, −7.435867928170042, −6.725848794030718, −5.475319891677333, −5.074754214421818, −3.827427426534318, −2.350890272668288, 0, 2.350890272668288, 3.827427426534318, 5.074754214421818, 5.475319891677333, 6.725848794030718, 7.435867928170042, 8.621807622472038, 9.987626462691324, 10.86614564966970, 11.45340338361601, 12.12691150722133, 12.98330226071648, 13.67165355575487, 14.73688343800399, 15.71634337737745, 15.99098091199238, 16.90557225798763, 17.85140258098576, 18.21374358987264, 19.40646700484665

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