Properties

Label 2-9800-1.1-c1-0-150
Degree $2$
Conductor $9800$
Sign $-1$
Analytic cond. $78.2533$
Root an. cond. $8.84609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 4·11-s − 2·13-s + 2·17-s − 4·19-s − 4·23-s − 2·29-s + 8·31-s − 6·37-s + 6·41-s + 8·43-s + 4·47-s − 6·53-s + 4·59-s + 2·61-s − 8·67-s − 6·73-s + 9·81-s − 16·83-s + 6·89-s − 14·97-s − 12·99-s − 6·101-s + 4·103-s + 14·109-s − 18·113-s + 6·117-s + ⋯
L(s)  = 1  − 9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.583·47-s − 0.824·53-s + 0.520·59-s + 0.256·61-s − 0.977·67-s − 0.702·73-s + 81-s − 1.75·83-s + 0.635·89-s − 1.42·97-s − 1.20·99-s − 0.597·101-s + 0.394·103-s + 1.34·109-s − 1.69·113-s + 0.554·117-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: yes
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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.36194372304734310007112788348, −6.52018592068616029985399260944, −6.04179762708888671493899602085, −5.38867194201806008960357176949, −4.41975332504676707087049724630, −3.91079757024853464020748642387, −2.95368945212330983062576397170, −2.25198268282814386124051356398, −1.20145405610619599855932818805, 0, 1.20145405610619599855932818805, 2.25198268282814386124051356398, 2.95368945212330983062576397170, 3.91079757024853464020748642387, 4.41975332504676707087049724630, 5.38867194201806008960357176949, 6.04179762708888671493899602085, 6.52018592068616029985399260944, 7.36194372304734310007112788348

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