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Properties

Label 2-1944-1944.1483-c0-0-0
Degree $2$
Conductor $1944$
Sign $0.770 - 0.637i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.627 − 0.778i)2-s + (−0.963 + 0.268i)3-s + (−0.211 + 0.977i)4-s + (0.813 + 0.581i)6-s + (0.893 − 0.448i)8-s + (0.856 − 0.516i)9-s + (0.638 + 1.33i)11-s + (−0.0581 − 0.998i)12-s + (−0.910 − 0.413i)16-s + (0.520 + 1.20i)17-s + (−0.939 − 0.342i)18-s + (−0.675 − 0.907i)19-s + (0.638 − 1.33i)22-s + (−0.740 + 0.672i)24-s + (−0.981 − 0.192i)25-s + ⋯
L(s)  = 1  + (−0.627 − 0.778i)2-s + (−0.963 + 0.268i)3-s + (−0.211 + 0.977i)4-s + (0.813 + 0.581i)6-s + (0.893 − 0.448i)8-s + (0.856 − 0.516i)9-s + (0.638 + 1.33i)11-s + (−0.0581 − 0.998i)12-s + (−0.910 − 0.413i)16-s + (0.520 + 1.20i)17-s + (−0.939 − 0.342i)18-s + (−0.675 − 0.907i)19-s + (0.638 − 1.33i)22-s + (−0.740 + 0.672i)24-s + (−0.981 − 0.192i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5410141466\)
\(L(\frac12)\) \(\approx\) \(0.5410141466\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.627 + 0.778i)T \)
3 \( 1 + (0.963 - 0.268i)T \)
good5 \( 1 + (0.981 + 0.192i)T^{2} \)
7 \( 1 + (-0.713 + 0.700i)T^{2} \)
11 \( 1 + (-0.638 - 1.33i)T + (-0.627 + 0.778i)T^{2} \)
13 \( 1 + (0.135 - 0.990i)T^{2} \)
17 \( 1 + (-0.520 - 1.20i)T + (-0.686 + 0.727i)T^{2} \)
19 \( 1 + (0.675 + 0.907i)T + (-0.286 + 0.957i)T^{2} \)
23 \( 1 + (0.963 - 0.268i)T^{2} \)
29 \( 1 + (0.999 + 0.0387i)T^{2} \)
31 \( 1 + (0.431 + 0.902i)T^{2} \)
37 \( 1 + (0.0581 - 0.998i)T^{2} \)
41 \( 1 + (-1.41 - 0.220i)T + (0.952 + 0.305i)T^{2} \)
43 \( 1 + (1.90 - 0.529i)T + (0.856 - 0.516i)T^{2} \)
47 \( 1 + (0.431 - 0.902i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-1.02 - 1.50i)T + (-0.360 + 0.932i)T^{2} \)
61 \( 1 + (0.910 - 0.413i)T^{2} \)
67 \( 1 + (-0.0359 - 1.85i)T + (-0.999 + 0.0387i)T^{2} \)
71 \( 1 + (0.993 - 0.116i)T^{2} \)
73 \( 1 + (-1.46 - 0.962i)T + (0.396 + 0.918i)T^{2} \)
79 \( 1 + (0.740 + 0.672i)T^{2} \)
83 \( 1 + (0.566 - 0.0886i)T + (0.952 - 0.305i)T^{2} \)
89 \( 1 + (0.0694 - 1.19i)T + (-0.993 - 0.116i)T^{2} \)
97 \( 1 + (-0.103 - 1.06i)T + (-0.981 + 0.192i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.757655475458807729295145526445, −8.917054463021377108207737885010, −8.000408736398143465120736551920, −7.08205410472566850517231812892, −6.48135310501094224148529464022, −5.33612225593277263658345824977, −4.27814320307697298953444129691, −3.88687093833908284675485396371, −2.32514767995185373546556767549, −1.28682223304872336343440880320, 0.62626866193901133151067265249, 1.86549367626299088872576871362, 3.62176852681491546425695315936, 4.74754002584649271013554807922, 5.63486352837797501986575095711, 6.11782816279882000213819846549, 6.86464627938105634877033817240, 7.70518995343305516529206832805, 8.343157387088167327899647946128, 9.305840970930460206141200507052

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