Properties

Label 2-280e2-1.1-c1-0-109
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  − 3·9-s − 6·11-s − 4·13-s − 7·17-s − 4·19-s + 23-s + 6·29-s + 3·31-s + 4·37-s − 9·41-s + 8·43-s − 11·47-s + 4·53-s − 10·59-s − 2·61-s + 8·67-s + 3·71-s − 2·73-s + 17·79-s + 9·81-s + 2·83-s − 7·89-s + 97-s + 18·99-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 9-s − 1.80·11-s − 1.10·13-s − 1.69·17-s − 0.917·19-s + 0.208·23-s + 1.11·29-s + 0.538·31-s + 0.657·37-s − 1.40·41-s + 1.21·43-s − 1.60·47-s + 0.549·53-s − 1.30·59-s − 0.256·61-s + 0.977·67-s + 0.356·71-s − 0.234·73-s + 1.91·79-s + 81-s + 0.219·83-s − 0.741·89-s + 0.101·97-s + 1.80·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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

−14.19505953159581, −13.72147526957911, −13.30978656834258, −12.71763594487225, −12.45459371491914, −11.68847054334714, −11.24914658393713, −10.75173590357560, −10.34876995753672, −9.824734945825509, −9.174205126254436, −8.561092184802181, −8.262863527357938, −7.717707201872697, −7.127018370591456, −6.413284878140485, −6.135061145344146, −5.258044919298143, −4.778939067863733, −4.596221565839898, −3.527436703039463, −2.797800154101559, −2.437679788844501, −1.989361832536794, −0.5805196833362792, 0, 0.5805196833362792, 1.989361832536794, 2.437679788844501, 2.797800154101559, 3.527436703039463, 4.596221565839898, 4.778939067863733, 5.258044919298143, 6.135061145344146, 6.413284878140485, 7.127018370591456, 7.717707201872697, 8.262863527357938, 8.561092184802181, 9.174205126254436, 9.824734945825509, 10.34876995753672, 10.75173590357560, 11.24914658393713, 11.68847054334714, 12.45459371491914, 12.71763594487225, 13.30978656834258, 13.72147526957911, 14.19505953159581

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