Properties

Label 2-280e2-1.1-c1-0-96
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s + 5·11-s + 5·17-s − 5·19-s + 6·23-s − 5·27-s − 4·29-s + 10·31-s + 5·33-s − 10·37-s − 5·41-s − 4·43-s + 8·47-s + 5·51-s − 10·53-s − 5·57-s − 10·61-s − 3·67-s + 6·69-s − 5·73-s − 10·79-s + 81-s − 83-s − 4·87-s + 9·89-s + 10·93-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s + 1.50·11-s + 1.21·17-s − 1.14·19-s + 1.25·23-s − 0.962·27-s − 0.742·29-s + 1.79·31-s + 0.870·33-s − 1.64·37-s − 0.780·41-s − 0.609·43-s + 1.16·47-s + 0.700·51-s − 1.37·53-s − 0.662·57-s − 1.28·61-s − 0.366·67-s + 0.722·69-s − 0.585·73-s − 1.12·79-s + 1/9·81-s − 0.109·83-s − 0.428·87-s + 0.953·89-s + 1.03·93-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

\(L(1)\) \(\approx\) \(2.971914384\)
\(L(\frac12)\) \(\approx\) \(2.971914384\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−14.10321877776972, −13.60590670017227, −13.15518837976282, −12.41836593403126, −11.98413005948395, −11.69841724713455, −11.00647944518017, −10.51609446885372, −9.946235137611713, −9.358562643303932, −8.868878299038073, −8.594591504687521, −8.013628613535364, −7.371134222994186, −6.830402165689733, −6.260094767374611, −5.836347073024434, −5.071226457678495, −4.514759580138508, −3.830649174329180, −3.237264459726666, −2.924681545126334, −1.915432555407681, −1.445765172295934, −0.5454263907676235, 0.5454263907676235, 1.445765172295934, 1.915432555407681, 2.924681545126334, 3.237264459726666, 3.830649174329180, 4.514759580138508, 5.071226457678495, 5.836347073024434, 6.260094767374611, 6.830402165689733, 7.371134222994186, 8.013628613535364, 8.594591504687521, 8.868878299038073, 9.358562643303932, 9.946235137611713, 10.51609446885372, 11.00647944518017, 11.69841724713455, 11.98413005948395, 12.41836593403126, 13.15518837976282, 13.60590670017227, 14.10321877776972

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