Properties

Label 2-9800-1.1-c1-0-122
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  + 3.28·3-s + 7.82·9-s + 5.82·11-s − 2.75·13-s − 2·17-s − 0.756·19-s + 0.533·23-s + 15.8·27-s − 0.823·29-s − 2.57·31-s + 19.1·33-s − 4.75·37-s − 9.06·39-s + 6.06·41-s − 0.710·43-s + 12.8·47-s − 6.57·51-s + 8.40·53-s − 2.48·57-s + 8·59-s + 9.40·61-s − 11.8·67-s + 1.75·69-s − 3.51·73-s − 9.51·79-s + 28.7·81-s + 6.71·83-s + ⋯
L(s)  = 1  + 1.89·3-s + 2.60·9-s + 1.75·11-s − 0.764·13-s − 0.485·17-s − 0.173·19-s + 0.111·23-s + 3.05·27-s − 0.152·29-s − 0.463·31-s + 3.33·33-s − 0.781·37-s − 1.45·39-s + 0.947·41-s − 0.108·43-s + 1.88·47-s − 0.921·51-s + 1.15·53-s − 0.329·57-s + 1.04·59-s + 1.20·61-s − 1.45·67-s + 0.211·69-s − 0.411·73-s − 1.07·79-s + 3.19·81-s + 0.736·83-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

\(L(1)\) \(\approx\) \(5.236791719\)
\(L(\frac12)\) \(\approx\) \(5.236791719\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.28T + 3T^{2} \)
11 \( 1 - 5.82T + 11T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 0.756T + 19T^{2} \)
23 \( 1 - 0.533T + 23T^{2} \)
29 \( 1 + 0.823T + 29T^{2} \)
31 \( 1 + 2.57T + 31T^{2} \)
37 \( 1 + 4.75T + 37T^{2} \)
41 \( 1 - 6.06T + 41T^{2} \)
43 \( 1 + 0.710T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 - 8.40T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 9.40T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 3.51T + 73T^{2} \)
79 \( 1 + 9.51T + 79T^{2} \)
83 \( 1 - 6.71T + 83T^{2} \)
89 \( 1 - 1.75T + 89T^{2} \)
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

−7.63184623267781307610240539823, −7.13631153378727979546336363640, −6.68122409044435126426704402157, −5.63908813239856305697544764078, −4.48781379162705647760970014303, −4.04607450602788549480953838947, −3.45440135599920505531046224731, −2.53947019993699431631820961406, −1.96905965839362431693077911145, −1.05057541153073013845727373776, 1.05057541153073013845727373776, 1.96905965839362431693077911145, 2.53947019993699431631820961406, 3.45440135599920505531046224731, 4.04607450602788549480953838947, 4.48781379162705647760970014303, 5.63908813239856305697544764078, 6.68122409044435126426704402157, 7.13631153378727979546336363640, 7.63184623267781307610240539823

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