Properties

Label 2-9800-1.1-c1-0-140
Degree $2$
Conductor $9800$
Sign $-1$
Analytic cond. $78.2533$
Root an. cond. $8.84609$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·3-s − 2.82·9-s + 0.828·11-s − 2·13-s + 7.65·17-s − 5.65·19-s + 5.58·23-s + 2.41·27-s − 7.82·29-s + 0.828·31-s − 0.343·33-s − 5.65·37-s + 0.828·39-s + 5.82·41-s + 6.89·43-s − 11.6·47-s − 3.17·51-s + 5.65·53-s + 2.34·57-s − 4·59-s + 6.65·61-s + 12.8·67-s − 2.31·69-s − 12·71-s − 3.65·73-s − 4·79-s + 7.48·81-s + ⋯
L(s)  = 1  − 0.239·3-s − 0.942·9-s + 0.249·11-s − 0.554·13-s + 1.85·17-s − 1.29·19-s + 1.16·23-s + 0.464·27-s − 1.45·29-s + 0.148·31-s − 0.0597·33-s − 0.929·37-s + 0.132·39-s + 0.910·41-s + 1.05·43-s − 1.70·47-s − 0.444·51-s + 0.777·53-s + 0.310·57-s − 0.520·59-s + 0.852·61-s + 1.57·67-s − 0.278·69-s − 1.42·71-s − 0.428·73-s − 0.450·79-s + 0.831·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9800\)    =    \(2^{3} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(78.2533\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 0.414T + 3T^{2} \)
11 \( 1 - 0.828T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 - 5.58T + 23T^{2} \)
29 \( 1 + 7.82T + 29T^{2} \)
31 \( 1 - 0.828T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 - 5.82T + 41T^{2} \)
43 \( 1 - 6.89T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 6.65T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 3.65T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 4.75T + 83T^{2} \)
89 \( 1 + 5.34T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.31190273626626586122409322862, −6.67355071855595058923641126825, −5.78023028878370388026184991397, −5.45970871883817431002797705811, −4.64586212396625846765261830298, −3.72359487845364375365054891267, −3.05556635382773581164498866073, −2.22158897405785341119485150975, −1.13761349690508980669908107473, 0, 1.13761349690508980669908107473, 2.22158897405785341119485150975, 3.05556635382773581164498866073, 3.72359487845364375365054891267, 4.64586212396625846765261830298, 5.45970871883817431002797705811, 5.78023028878370388026184991397, 6.67355071855595058923641126825, 7.31190273626626586122409322862

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