Properties

Label 2-9800-1.1-c1-0-46
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 $0$

Origins

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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: \(0\)
Selberg data: \((2,\ 9800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.829174579\)
\(L(\frac12)\) \(\approx\) \(1.829174579\)
\(L(\frac{3}{2})\) 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 \( 1 - 6T + 97T^{2} \)
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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

−7.52945608121675176218474087850, −7.10949935423580399176141247337, −6.29959618861199613365053970508, −5.59783094015890141124388957858, −5.02707574767432755245329020092, −4.06002410212405052187329449350, −3.41927647368554169745515583158, −2.62429489880858077805972665759, −1.81667089439040978754705944090, −0.62305154198281748890644989144, 0.62305154198281748890644989144, 1.81667089439040978754705944090, 2.62429489880858077805972665759, 3.41927647368554169745515583158, 4.06002410212405052187329449350, 5.02707574767432755245329020092, 5.59783094015890141124388957858, 6.29959618861199613365053970508, 7.10949935423580399176141247337, 7.52945608121675176218474087850

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