Properties

Label 2-9800-1.1-c1-0-94
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  − 3·3-s + 6·9-s − 11-s − 2·13-s − 3·17-s + 5·19-s + 3·23-s − 9·27-s − 6·29-s − 31-s + 3·33-s + 5·37-s + 6·39-s − 10·41-s + 4·43-s − 47-s + 9·51-s + 9·53-s − 15·57-s + 3·59-s + 3·61-s − 11·67-s − 9·69-s + 16·71-s − 7·73-s − 11·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s − 0.301·11-s − 0.554·13-s − 0.727·17-s + 1.14·19-s + 0.625·23-s − 1.73·27-s − 1.11·29-s − 0.179·31-s + 0.522·33-s + 0.821·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s − 0.145·47-s + 1.26·51-s + 1.23·53-s − 1.98·57-s + 0.390·59-s + 0.384·61-s − 1.34·67-s − 1.08·69-s + 1.89·71-s − 0.819·73-s − 1.23·79-s + 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: 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 \( 1 + p T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.08339889480932922472176429884, −6.72611475509514317854444547667, −5.71339868968163172971506673055, −5.45956601864675439321957310826, −4.74414087860660108859448707952, −4.09717026818513344858517515159, −3.05130381553721572926622034755, −1.95351204366671363904356474654, −0.939899853876546082418835939449, 0, 0.939899853876546082418835939449, 1.95351204366671363904356474654, 3.05130381553721572926622034755, 4.09717026818513344858517515159, 4.74414087860660108859448707952, 5.45956601864675439321957310826, 5.71339868968163172971506673055, 6.72611475509514317854444547667, 7.08339889480932922472176429884

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