Properties

Label 2-1400-1.1-c1-0-19
Degree $2$
Conductor $1400$
Sign $-1$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
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  − 2·3-s + 7-s + 9-s + 11-s − 4·13-s + 6·19-s − 2·21-s − 3·23-s + 4·27-s − 3·29-s − 2·33-s − 9·37-s + 8·39-s + 2·41-s − 9·43-s + 6·47-s + 49-s − 6·53-s − 12·57-s + 8·59-s − 10·61-s + 63-s + 67-s + 6·69-s − 7·71-s + 2·73-s + 77-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1.37·19-s − 0.436·21-s − 0.625·23-s + 0.769·27-s − 0.557·29-s − 0.348·33-s − 1.47·37-s + 1.28·39-s + 0.312·41-s − 1.37·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.58·57-s + 1.04·59-s − 1.28·61-s + 0.125·63-s + 0.122·67-s + 0.722·69-s − 0.830·71-s + 0.234·73-s + 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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

−9.328240328959893836820488605573, −8.291638824354553749832147975540, −7.36093716429216483962761853710, −6.71609133607624229944036178184, −5.60555227321757038670706914438, −5.20829029475748356758143176483, −4.22360783809719513353208535534, −2.94349967687063095785099679694, −1.51296624879668184664231800532, 0, 1.51296624879668184664231800532, 2.94349967687063095785099679694, 4.22360783809719513353208535534, 5.20829029475748356758143176483, 5.60555227321757038670706914438, 6.71609133607624229944036178184, 7.36093716429216483962761853710, 8.291638824354553749832147975540, 9.328240328959893836820488605573

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