↑

Properties

Label 2-280e2-1.1-c1-0-279
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 6·9-s − 5·11-s + 3·13-s − 17-s − 6·19-s + 6·23-s + 9·27-s + 9·29-s − 4·31-s − 15·33-s + 2·37-s + 9·39-s + 4·41-s − 10·43-s + 47-s − 3·51-s + 4·53-s − 18·57-s + 8·59-s − 8·61-s − 12·67-s + 18·69-s − 8·71-s + 2·73-s − 13·79-s + 9·81-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s − 1.50·11-s + 0.832·13-s − 0.242·17-s − 1.37·19-s + 1.25·23-s + 1.73·27-s + 1.67·29-s − 0.718·31-s − 2.61·33-s + 0.328·37-s + 1.44·39-s + 0.624·41-s − 1.52·43-s + 0.145·47-s − 0.420·51-s + 0.549·53-s − 2.38·57-s + 1.04·59-s − 1.02·61-s − 1.46·67-s + 2.16·69-s − 0.949·71-s + 0.234·73-s − 1.46·79-s + 81-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 78400,\ (\ :1/2),\ -1)\)

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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.32978420650380, −13.61447684961151, −13.28125525779597, −13.09871655778596, −12.54832639408093, −11.90097485495574, −10.99238841334146, −10.71937520960176, −10.20230362107169, −9.721294707105335, −8.970126167081417, −8.643631374417443, −8.301260318924382, −7.851863102060442, −7.150850870410701, −6.810556703972446, −6.029822845604533, −5.359034126117312, −4.558612685824407, −4.275298185612727, −3.454506431470042, −2.851739035616002, −2.618196704342212, −1.839116985275708, −1.163434159799124, 0, 1.163434159799124, 1.839116985275708, 2.618196704342212, 2.851739035616002, 3.454506431470042, 4.275298185612727, 4.558612685824407, 5.359034126117312, 6.029822845604533, 6.810556703972446, 7.150850870410701, 7.851863102060442, 8.301260318924382, 8.643631374417443, 8.970126167081417, 9.721294707105335, 10.20230362107169, 10.71937520960176, 10.99238841334146, 11.90097485495574, 12.54832639408093, 13.09871655778596, 13.28125525779597, 13.61447684961151, 14.32978420650380

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