↑

Properties

Label 2-1944-216.115-c0-0-2
Degree $2$
Conductor $1944$
Sign $0.727 + 0.686i$
Analytic cond. $0.970182$
Root an. cond. $0.984978$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.500 − 0.866i)8-s + (−0.0603 + 0.342i)11-s + (0.173 − 0.984i)16-s + (0.766 + 1.32i)17-s + (0.939 − 1.62i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)25-s + (−0.173 − 0.984i)32-s + (1.17 + 0.984i)34-s + (0.326 − 1.85i)38-s + (−1.76 − 0.642i)41-s + (0.266 − 1.50i)43-s + (0.173 + 0.300i)44-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.500 − 0.866i)8-s + (−0.0603 + 0.342i)11-s + (0.173 − 0.984i)16-s + (0.766 + 1.32i)17-s + (0.939 − 1.62i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)25-s + (−0.173 − 0.984i)32-s + (1.17 + 0.984i)34-s + (0.326 − 1.85i)38-s + (−1.76 − 0.642i)41-s + (0.266 − 1.50i)43-s + (0.173 + 0.300i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.097321447\)
\(L(\frac12)\) \(\approx\) \(2.097321447\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
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

−9.479219542787839281147284135558, −8.519536159402588829981143444846, −7.42191464728331298599511698794, −6.91909225466686203560035154159, −5.82222992541536538892275054857, −5.29808710211540617850403000830, −4.28810742579248703809278881907, −3.50714251151926768098134352075, −2.52106935241553668683576681407, −1.39076086364621085132858389440, 1.66005251536660642911708995972, 3.00509583650460613396024623902, 3.60063365439144641765397653015, 4.71409941221615438235013347329, 5.47700992218194699331792128552, 6.12020759576918360398589881631, 7.06810651675586710375967179353, 7.82284603543932532878184446354, 8.339363714950407144214335248047, 9.627613711237732722321916016285

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