↑

Properties

Label 2-1200-12.11-c1-0-30
Degree $2$
Conductor $1200$
Sign $i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − 1.73i·7-s − 2.99·9-s − 5·13-s − 8.66i·19-s + 2.99·21-s − 5.19i·27-s − 8.66i·31-s − 10·37-s − 8.66i·39-s + 12.1i·43-s + 4·49-s + 15·57-s − 13·61-s + 5.19i·63-s + ⋯
L(s)  = 1  + 0.999i·3-s − 0.654i·7-s − 0.999·9-s − 1.38·13-s − 1.98i·19-s + 0.654·21-s − 0.999i·27-s − 1.55i·31-s − 1.64·37-s − 1.38i·39-s + 1.84i·43-s + 0.571·49-s + 1.98·57-s − 1.66·61-s + 0.654i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6886185137\)
\(L(\frac12)\) \(\approx\) \(0.6886185137\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 8.66iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 15.5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 17.3iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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

−9.514800567329425139879780081743, −9.043489449674298700228258812157, −7.88082001191237553437677673844, −7.14631336816116691728203265300, −6.13240513287295041899967070526, −4.90037123011317828003612267126, −4.57987201727591945190672039310, −3.36459917526947416790204583042, −2.39643796912090144476165201634, −0.27993875093401309848081327225, 1.58955996452107888189024399667, 2.52121662760051160279188028485, 3.63911607407508923724077599113, 5.16068530292061109012537724793, 5.70991996374891561132490399374, 6.78317395242220721289247436099, 7.41569766591813217294246981853, 8.305282279399175030971269834257, 8.919020044526602431846239970269, 9.988602571823711885986141381344

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