Properties

Label 2-490-1.1-c1-0-6
Degree $2$
Conductor $490$
Sign $1$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s − 3·9-s + 10-s + 4·11-s + 6·13-s + 16-s − 2·17-s − 3·18-s + 20-s + 4·22-s + 25-s + 6·26-s + 6·29-s − 8·31-s + 32-s − 2·34-s − 3·36-s − 10·37-s + 40-s − 2·41-s + 4·43-s + 4·44-s − 3·45-s − 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 9-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.64·37-s + 0.158·40-s − 0.312·41-s + 0.609·43-s + 0.603·44-s − 0.447·45-s − 1.16·47-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.369641784\)
\(L(\frac12)\) \(\approx\) \(2.369641784\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−11.21735898278523397026755152007, −10.27331483782898950038120178987, −8.943481127314352432901190185250, −8.517395214779930601558675802260, −6.94587741637952917868946998696, −6.19981958164339530240516278001, −5.43748531590055688874527891076, −4.08647900838425361258038365025, −3.14888451818473432527395410089, −1.60311130730382042811195419884, 1.60311130730382042811195419884, 3.14888451818473432527395410089, 4.08647900838425361258038365025, 5.43748531590055688874527891076, 6.19981958164339530240516278001, 6.94587741637952917868946998696, 8.517395214779930601558675802260, 8.943481127314352432901190185250, 10.27331483782898950038120178987, 11.21735898278523397026755152007

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