↑

Properties

Label 2-1944-1944.1123-c0-0-0
Degree $2$
Conductor $1944$
Sign $-0.280 - 0.959i$
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  + (−0.963 + 0.268i)2-s + (−0.740 + 0.672i)3-s + (0.856 − 0.516i)4-s + (0.533 − 0.845i)6-s + (−0.686 + 0.727i)8-s + (0.0968 − 0.995i)9-s + (−0.00524 + 0.0384i)11-s + (−0.286 + 0.957i)12-s + (0.466 − 0.884i)16-s + (−1.01 − 0.507i)17-s + (0.173 + 0.984i)18-s + (−0.107 + 1.84i)19-s + (−0.00524 − 0.0384i)22-s + (0.0193 − 0.999i)24-s + (0.996 + 0.0774i)25-s + ⋯
L(s)  = 1  + (−0.963 + 0.268i)2-s + (−0.740 + 0.672i)3-s + (0.856 − 0.516i)4-s + (0.533 − 0.845i)6-s + (−0.686 + 0.727i)8-s + (0.0968 − 0.995i)9-s + (−0.00524 + 0.0384i)11-s + (−0.286 + 0.957i)12-s + (0.466 − 0.884i)16-s + (−1.01 − 0.507i)17-s + (0.173 + 0.984i)18-s + (−0.107 + 1.84i)19-s + (−0.00524 − 0.0384i)22-s + (0.0193 − 0.999i)24-s + (0.996 + 0.0774i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.280 - 0.959i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.280 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4810221668\)
\(L(\frac12)\) \(\approx\) \(0.4810221668\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.963 - 0.268i)T \)
3 \( 1 + (0.740 - 0.672i)T \)
good5 \( 1 + (-0.996 - 0.0774i)T^{2} \)
7 \( 1 + (-0.952 + 0.305i)T^{2} \)
11 \( 1 + (0.00524 - 0.0384i)T + (-0.963 - 0.268i)T^{2} \)
13 \( 1 + (0.360 - 0.932i)T^{2} \)
17 \( 1 + (1.01 + 0.507i)T + (0.597 + 0.802i)T^{2} \)
19 \( 1 + (0.107 - 1.84i)T + (-0.993 - 0.116i)T^{2} \)
23 \( 1 + (0.740 - 0.672i)T^{2} \)
29 \( 1 + (-0.323 + 0.946i)T^{2} \)
31 \( 1 + (0.135 - 0.990i)T^{2} \)
37 \( 1 + (0.286 + 0.957i)T^{2} \)
41 \( 1 + (-0.475 - 1.84i)T + (-0.875 + 0.483i)T^{2} \)
43 \( 1 + (0.369 - 0.335i)T + (0.0968 - 0.995i)T^{2} \)
47 \( 1 + (0.135 + 0.990i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.862 - 0.352i)T + (0.713 + 0.700i)T^{2} \)
61 \( 1 + (-0.466 - 0.884i)T^{2} \)
67 \( 1 + (-1.60 - 1.14i)T + (0.323 + 0.946i)T^{2} \)
71 \( 1 + (0.835 + 0.549i)T^{2} \)
73 \( 1 + (1.77 + 0.419i)T + (0.893 + 0.448i)T^{2} \)
79 \( 1 + (-0.0193 - 0.999i)T^{2} \)
83 \( 1 + (0.495 - 1.92i)T + (-0.875 - 0.483i)T^{2} \)
89 \( 1 + (-0.0333 - 0.111i)T + (-0.835 + 0.549i)T^{2} \)
97 \( 1 + (1.31 - 0.0509i)T + (0.996 - 0.0774i)T^{2} \)
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

−9.758465447675810524536112889733, −8.870214338561568393482735195167, −8.233843079863372369077270697433, −7.19674395619581281054993789624, −6.48567336350288134273453107417, −5.78208365793057620411086628079, −4.94436284694071721909217485908, −3.90905284255189062622359680548, −2.67562056617244930908618414758, −1.23610383306555588375937190546, 0.58260662461013936422238210551, 1.94997569881273565515066673343, 2.77693553167777207949090835984, 4.22070397338988107217762005993, 5.27072283621709791731417888851, 6.30449971915294478900543820557, 6.94861698448806061460409388246, 7.43109117190020257127483318292, 8.611009138154611994281970400333, 8.890161343346281336467858625808

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