Properties

Degree $2$
Conductor $9800$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s − 2·17-s + 2·19-s − 8·23-s − 4·27-s + 2·29-s − 4·31-s + 6·37-s + 2·41-s − 8·43-s − 4·47-s − 4·51-s + 10·53-s + 4·57-s − 6·59-s − 4·61-s + 12·67-s − 16·69-s − 14·73-s − 8·79-s − 11·81-s + 6·83-s + 4·87-s − 10·89-s − 8·93-s − 2·97-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 0.485·17-s + 0.458·19-s − 1.66·23-s − 0.769·27-s + 0.371·29-s − 0.718·31-s + 0.986·37-s + 0.312·41-s − 1.21·43-s − 0.583·47-s − 0.560·51-s + 1.37·53-s + 0.529·57-s − 0.781·59-s − 0.512·61-s + 1.46·67-s − 1.92·69-s − 1.63·73-s − 0.900·79-s − 1.22·81-s + 0.658·83-s + 0.428·87-s − 1.05·89-s − 0.829·93-s − 0.203·97-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: $-1$
Motivic weight: \(1\)
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.55983630513273816716993425089, −6.74223267938098275326169027357, −5.99839169264097337665023383664, −5.29351484382949358954324263039, −4.30166580613394842173090466733, −3.77456624676684602640293661127, −2.94603065379908432306815937132, −2.29863418158137296206682798774, −1.48341039870823931767409571367, 0, 1.48341039870823931767409571367, 2.29863418158137296206682798774, 2.94603065379908432306815937132, 3.77456624676684602640293661127, 4.30166580613394842173090466733, 5.29351484382949358954324263039, 5.99839169264097337665023383664, 6.74223267938098275326169027357, 7.55983630513273816716993425089

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