Properties

Label 2-490-1.1-c1-0-9
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 + 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: $1$
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

\(L(1)\) \(\approx\) \(3.071649458\)
\(L(\frac12)\) \(\approx\) \(3.071649458\)
\(L(\frac{3}{2})\) 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

−10.78524842246065370835052081258, −10.18076940563670721337529296568, −9.102066376137223109803910203953, −8.118957846682560780166935032553, −7.54815335398458076593938530060, −6.10181652811109734263548661708, −5.34673915974432534760216274251, −3.93281458879405714793180621994, −2.99518962250547346483426263862, −1.97892037150738648722925117261, 1.97892037150738648722925117261, 2.99518962250547346483426263862, 3.93281458879405714793180621994, 5.34673915974432534760216274251, 6.10181652811109734263548661708, 7.54815335398458076593938530060, 8.118957846682560780166935032553, 9.102066376137223109803910203953, 10.18076940563670721337529296568, 10.78524842246065370835052081258

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