Properties

Label 2-490-1.1-c1-0-8
Degree $2$
Conductor $490$
Sign $-1$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.585·3-s + 4-s − 5-s + 0.585·6-s − 8-s − 2.65·9-s + 10-s + 4.82·11-s − 0.585·12-s − 0.828·13-s + 0.585·15-s + 16-s − 5.41·17-s + 2.65·18-s − 3.41·19-s − 20-s − 4.82·22-s − 6.82·23-s + 0.585·24-s + 25-s + 0.828·26-s + 3.31·27-s + 0.828·29-s − 0.585·30-s − 2.82·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.338·3-s + 0.5·4-s − 0.447·5-s + 0.239·6-s − 0.353·8-s − 0.885·9-s + 0.316·10-s + 1.45·11-s − 0.169·12-s − 0.229·13-s + 0.151·15-s + 0.250·16-s − 1.31·17-s + 0.626·18-s − 0.783·19-s − 0.223·20-s − 1.02·22-s − 1.42·23-s + 0.119·24-s + 0.200·25-s + 0.162·26-s + 0.637·27-s + 0.153·29-s − 0.106·30-s − 0.508·31-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: $-1$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (1, \cdot )$
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 \( 1 + 0.585T + 3T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 + 5.41T + 17T^{2} \)
19 \( 1 + 3.41T + 19T^{2} \)
23 \( 1 + 6.82T + 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 3.17T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + 9.65T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 6.58T + 73T^{2} \)
79 \( 1 + 1.17T + 79T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−10.58610337582269844470310669795, −9.515257255811814933643830066743, −8.710215133886423869714400237578, −8.027505829786312290029656815937, −6.67706129260928183873608072134, −6.24121911688125710111233764299, −4.72405063829918254237162612210, −3.53228460490233816266070308549, −1.95349450908375665927514463418, 0, 1.95349450908375665927514463418, 3.53228460490233816266070308549, 4.72405063829918254237162612210, 6.24121911688125710111233764299, 6.67706129260928183873608072134, 8.027505829786312290029656815937, 8.710215133886423869714400237578, 9.515257255811814933643830066743, 10.58610337582269844470310669795

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