Properties

Label 2-280e2-1.1-c1-0-14
Degree $2$
Conductor $78400$
Sign $1$
Analytic cond. $626.027$
Root an. cond. $25.0205$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s − 3·11-s + 7·13-s − 5·17-s − 4·19-s − 4·23-s − 5·27-s + 5·29-s − 2·31-s − 3·33-s + 7·39-s − 6·41-s − 8·43-s − 9·47-s − 5·51-s − 4·57-s − 12·59-s − 6·61-s − 16·67-s − 4·69-s + 4·71-s − 14·73-s + 9·79-s + 81-s − 8·83-s + 5·87-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s − 0.904·11-s + 1.94·13-s − 1.21·17-s − 0.917·19-s − 0.834·23-s − 0.962·27-s + 0.928·29-s − 0.359·31-s − 0.522·33-s + 1.12·39-s − 0.937·41-s − 1.21·43-s − 1.31·47-s − 0.700·51-s − 0.529·57-s − 1.56·59-s − 0.768·61-s − 1.95·67-s − 0.481·69-s + 0.474·71-s − 1.63·73-s + 1.01·79-s + 1/9·81-s − 0.878·83-s + 0.536·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(626.027\)
Root analytic conductor: \(25.0205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{78400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 78400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8879340420\)
\(L(\frac12)\) \(\approx\) \(0.8879340420\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−13.76911313361673, −13.62326533134665, −13.16653169382034, −12.75099036172529, −11.89049866365017, −11.53153592395431, −10.97527712631393, −10.48433225559440, −10.21063555206828, −9.216403354241674, −8.909157562720408, −8.380849600104089, −8.142237977783899, −7.558022331302455, −6.594857268963011, −6.335034327106272, −5.848658231683870, −5.096350654474523, −4.474236557810685, −3.916301991397829, −3.174865401276885, −2.879123058470611, −1.919936929904904, −1.588910702539144, −0.2757621890436199, 0.2757621890436199, 1.588910702539144, 1.919936929904904, 2.879123058470611, 3.174865401276885, 3.916301991397829, 4.474236557810685, 5.096350654474523, 5.848658231683870, 6.335034327106272, 6.594857268963011, 7.558022331302455, 8.142237977783899, 8.380849600104089, 8.909157562720408, 9.216403354241674, 10.21063555206828, 10.48433225559440, 10.97527712631393, 11.53153592395431, 11.89049866365017, 12.75099036172529, 13.16653169382034, 13.62326533134665, 13.76911313361673

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