Properties

Label 2-9800-1.1-c1-0-130
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.12·3-s + 6.76·9-s + 2.48·11-s + 4.15·13-s − 5.76·17-s + 1.60·19-s + 7.28·23-s − 11.7·27-s + 1.45·29-s + 2.24·31-s − 7.76·33-s − 6·37-s − 12.9·39-s − 11.2·41-s − 5.28·43-s − 3.45·47-s + 18.0·51-s − 9.21·53-s − 5.03·57-s + 5.92·59-s − 5.35·61-s − 7.52·67-s − 22.7·69-s − 4.24·71-s − 7.28·73-s + 16.9·79-s + 16.4·81-s + ⋯
L(s)  = 1  − 1.80·3-s + 2.25·9-s + 0.749·11-s + 1.15·13-s − 1.39·17-s + 0.369·19-s + 1.51·23-s − 2.26·27-s + 0.270·29-s + 0.404·31-s − 1.35·33-s − 0.986·37-s − 2.07·39-s − 1.76·41-s − 0.805·43-s − 0.503·47-s + 2.52·51-s − 1.26·53-s − 0.666·57-s + 0.770·59-s − 0.686·61-s − 0.919·67-s − 2.73·69-s − 0.504·71-s − 0.852·73-s + 1.91·79-s + 1.82·81-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: Trivial
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 \( 1 + 3.12T + 3T^{2} \)
11 \( 1 - 2.48T + 11T^{2} \)
13 \( 1 - 4.15T + 13T^{2} \)
17 \( 1 + 5.76T + 17T^{2} \)
19 \( 1 - 1.60T + 19T^{2} \)
23 \( 1 - 7.28T + 23T^{2} \)
29 \( 1 - 1.45T + 29T^{2} \)
31 \( 1 - 2.24T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 5.28T + 43T^{2} \)
47 \( 1 + 3.45T + 47T^{2} \)
53 \( 1 + 9.21T + 53T^{2} \)
59 \( 1 - 5.92T + 59T^{2} \)
61 \( 1 + 5.35T + 61T^{2} \)
67 \( 1 + 7.52T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 7.28T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 2.73T + 97T^{2} \)
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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

−6.83060634260935490854640187992, −6.63086223981356413318771171877, −6.14602813777282541664424761077, −5.12498263918387015289198724300, −4.88061085036576032263737559258, −3.99542494216663733990644151862, −3.21976239501058502223547587228, −1.73670139120075347509436197335, −1.09446416799895602715214706682, 0, 1.09446416799895602715214706682, 1.73670139120075347509436197335, 3.21976239501058502223547587228, 3.99542494216663733990644151862, 4.88061085036576032263737559258, 5.12498263918387015289198724300, 6.14602813777282541664424761077, 6.63086223981356413318771171877, 6.83060634260935490854640187992

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