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Properties

Label 2-1944-1944.139-c0-0-0
Degree $2$
Conductor $1944$
Sign $0.971 - 0.236i$
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  + (−0.211 + 0.977i)2-s + (0.856 − 0.516i)3-s + (−0.910 − 0.413i)4-s + (0.323 + 0.946i)6-s + (0.597 − 0.802i)8-s + (0.466 − 0.884i)9-s + (−0.121 + 0.150i)11-s + (−0.993 + 0.116i)12-s + (0.657 + 0.753i)16-s + (0.185 − 0.196i)17-s + (0.766 + 0.642i)18-s + (0.206 − 0.690i)19-s + (−0.121 − 0.150i)22-s + (0.0968 − 0.995i)24-s + (0.925 + 0.378i)25-s + ⋯
L(s)  = 1  + (−0.211 + 0.977i)2-s + (0.856 − 0.516i)3-s + (−0.910 − 0.413i)4-s + (0.323 + 0.946i)6-s + (0.597 − 0.802i)8-s + (0.466 − 0.884i)9-s + (−0.121 + 0.150i)11-s + (−0.993 + 0.116i)12-s + (0.657 + 0.753i)16-s + (0.185 − 0.196i)17-s + (0.766 + 0.642i)18-s + (0.206 − 0.690i)19-s + (−0.121 − 0.150i)22-s + (0.0968 − 0.995i)24-s + (0.925 + 0.378i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.971 - 0.236i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.971 - 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.294755343\)
\(L(\frac12)\) \(\approx\) \(1.294755343\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.211 - 0.977i)T \)
3 \( 1 + (-0.856 + 0.516i)T \)
good5 \( 1 + (-0.925 - 0.378i)T^{2} \)
7 \( 1 + (-0.0193 + 0.999i)T^{2} \)
11 \( 1 + (0.121 - 0.150i)T + (-0.211 - 0.977i)T^{2} \)
13 \( 1 + (0.963 + 0.268i)T^{2} \)
17 \( 1 + (-0.185 + 0.196i)T + (-0.0581 - 0.998i)T^{2} \)
19 \( 1 + (-0.206 + 0.690i)T + (-0.835 - 0.549i)T^{2} \)
23 \( 1 + (-0.856 + 0.516i)T^{2} \)
29 \( 1 + (-0.996 - 0.0774i)T^{2} \)
31 \( 1 + (0.627 - 0.778i)T^{2} \)
37 \( 1 + (0.993 + 0.116i)T^{2} \)
41 \( 1 + (-0.0369 - 0.0118i)T + (0.813 + 0.581i)T^{2} \)
43 \( 1 + (-1.63 + 0.984i)T + (0.466 - 0.884i)T^{2} \)
47 \( 1 + (0.627 + 0.778i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (0.473 - 1.22i)T + (-0.740 - 0.672i)T^{2} \)
61 \( 1 + (-0.657 + 0.753i)T^{2} \)
67 \( 1 + (1.42 - 0.0553i)T + (0.996 - 0.0774i)T^{2} \)
71 \( 1 + (-0.973 + 0.230i)T^{2} \)
73 \( 1 + (-0.422 - 0.979i)T + (-0.686 + 0.727i)T^{2} \)
79 \( 1 + (-0.0968 - 0.995i)T^{2} \)
83 \( 1 + (1.59 - 0.510i)T + (0.813 - 0.581i)T^{2} \)
89 \( 1 + (-0.569 - 0.0665i)T + (0.973 + 0.230i)T^{2} \)
97 \( 1 + (-0.846 + 0.166i)T + (0.925 - 0.378i)T^{2} \)
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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

−8.974492426706736615814093080467, −8.741818021726120637184270711510, −7.64298476139386752882442302490, −7.26229713749466398130826745236, −6.49142212666470696520659920659, −5.55740691169218195827639591940, −4.60709099521921283883800001487, −3.65111000336474939181273950025, −2.53243061425713457051487626247, −1.10079551765817248289829685509, 1.45203730918944330848706985374, 2.60375338526240354159266552574, 3.31244621271047309938737141337, 4.22242825425522412586606216449, 4.92222636039356417367332641169, 6.01852955203955201890051993275, 7.44724886061166239756561078533, 8.007068314704283314608690764070, 8.801080326756803030802838731182, 9.366065916567246857748769523480

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