Properties

Label 2-9800-1.1-c1-0-73
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  − 2.58·3-s + 3.67·9-s + 1.67·11-s + 4.84·13-s − 2·17-s + 6.84·19-s + 2.26·23-s − 1.75·27-s + 3.32·29-s + 9.16·31-s − 4.33·33-s + 2.84·37-s − 12.5·39-s + 9.52·41-s − 6.58·43-s + 12.2·47-s + 5.16·51-s − 7.49·53-s − 17.6·57-s + 8·59-s − 6.49·61-s + 5.75·67-s − 5.84·69-s + 11.6·73-s + 5.69·79-s − 6.50·81-s + 12.5·83-s + ⋯
L(s)  = 1  − 1.49·3-s + 1.22·9-s + 0.506·11-s + 1.34·13-s − 0.485·17-s + 1.57·19-s + 0.471·23-s − 0.337·27-s + 0.616·29-s + 1.64·31-s − 0.754·33-s + 0.467·37-s − 2.00·39-s + 1.48·41-s − 1.00·43-s + 1.78·47-s + 0.723·51-s − 1.02·53-s − 2.34·57-s + 1.04·59-s − 0.830·61-s + 0.702·67-s − 0.703·69-s + 1.36·73-s + 0.640·79-s − 0.722·81-s + 1.38·83-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

\(L(1)\) \(\approx\) \(1.624849254\)
\(L(\frac12)\) \(\approx\) \(1.624849254\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.58T + 3T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 - 4.84T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 6.84T + 19T^{2} \)
23 \( 1 - 2.26T + 23T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 - 9.16T + 31T^{2} \)
37 \( 1 - 2.84T + 37T^{2} \)
41 \( 1 - 9.52T + 41T^{2} \)
43 \( 1 + 6.58T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + 7.49T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 6.49T + 61T^{2} \)
67 \( 1 - 5.75T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 5.69T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 5.84T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−7.54045709955531533065587622466, −6.64420061917747104628153655563, −6.36586205418435968618229585060, −5.67872403750231755162011090531, −5.03586168565196019868266340216, −4.35695122352475839599642793456, −3.58064795246866162811108579113, −2.61339436289299654333652692531, −1.19846087975663751007791416174, −0.816572692592439060531618931317, 0.816572692592439060531618931317, 1.19846087975663751007791416174, 2.61339436289299654333652692531, 3.58064795246866162811108579113, 4.35695122352475839599642793456, 5.03586168565196019868266340216, 5.67872403750231755162011090531, 6.36586205418435968618229585060, 6.64420061917747104628153655563, 7.54045709955531533065587622466

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