Properties

Label 2-280e2-1.1-c1-0-31
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  − 2·3-s + 9-s − 11-s + 2·13-s + 4·17-s − 5·23-s + 4·27-s + 3·29-s + 10·31-s + 2·33-s − 5·37-s − 4·39-s − 10·41-s − 5·43-s + 4·47-s − 8·51-s − 10·53-s − 10·59-s − 10·61-s + 5·67-s + 10·69-s − 3·71-s − 10·73-s − 13·79-s − 11·81-s − 10·83-s − 6·87-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.970·17-s − 1.04·23-s + 0.769·27-s + 0.557·29-s + 1.79·31-s + 0.348·33-s − 0.821·37-s − 0.640·39-s − 1.56·41-s − 0.762·43-s + 0.583·47-s − 1.12·51-s − 1.37·53-s − 1.30·59-s − 1.28·61-s + 0.610·67-s + 1.20·69-s − 0.356·71-s − 1.17·73-s − 1.46·79-s − 1.22·81-s − 1.09·83-s − 0.643·87-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

\(L(1)\) \(\approx\) \(0.7875716288\)
\(L(\frac12)\) \(\approx\) \(0.7875716288\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−14.01462426975300, −13.53969966052877, −12.96807743292898, −12.25546795070183, −11.96793547612827, −11.73034279694982, −11.01344680573539, −10.47979439827697, −10.18284116878292, −9.729893584423941, −8.892210249679330, −8.339755302159326, −7.989891176275311, −7.287290090059626, −6.606189369666717, −6.205510323673164, −5.790438663084387, −5.167282922799998, −4.696142748494462, −4.129556913381725, −3.187193078423250, −2.924375461837352, −1.760736142794549, −1.239391685092634, −0.3333029434277437, 0.3333029434277437, 1.239391685092634, 1.760736142794549, 2.924375461837352, 3.187193078423250, 4.129556913381725, 4.696142748494462, 5.167282922799998, 5.790438663084387, 6.205510323673164, 6.606189369666717, 7.287290090059626, 7.989891176275311, 8.339755302159326, 8.892210249679330, 9.729893584423941, 10.18284116878292, 10.47979439827697, 11.01344680573539, 11.73034279694982, 11.96793547612827, 12.25546795070183, 12.96807743292898, 13.53969966052877, 14.01462426975300

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