Properties

Label 2-9800-1.1-c1-0-177
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  + 2·3-s + 9-s − 11-s + 3·13-s + 2·17-s − 5·19-s − 7·23-s − 4·27-s − 6·29-s + 4·31-s − 2·33-s + 5·37-s + 6·39-s − 5·41-s − 6·43-s + 9·47-s + 4·51-s − 11·53-s − 10·57-s + 8·59-s − 12·61-s + 4·67-s − 14·69-s − 4·71-s − 12·73-s + 14·79-s − 11·81-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.485·17-s − 1.14·19-s − 1.45·23-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s + 0.821·37-s + 0.960·39-s − 0.780·41-s − 0.914·43-s + 1.31·47-s + 0.560·51-s − 1.51·53-s − 1.32·57-s + 1.04·59-s − 1.53·61-s + 0.488·67-s − 1.68·69-s − 0.474·71-s − 1.40·73-s + 1.57·79-s − 1.22·81-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−7.70748305988922431541472995926, −6.60979693662959110686763825647, −6.05805678840226096811373754318, −5.32324687170290190938299814191, −4.24949143485096388328386161889, −3.78230390358448701545296133213, −3.00108364896703802275334761778, −2.24268542827022201375721187011, −1.50511538093886971774076910779, 0, 1.50511538093886971774076910779, 2.24268542827022201375721187011, 3.00108364896703802275334761778, 3.78230390358448701545296133213, 4.24949143485096388328386161889, 5.32324687170290190938299814191, 6.05805678840226096811373754318, 6.60979693662959110686763825647, 7.70748305988922431541472995926

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