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Properties

Label 2-1944-1944.1651-c0-0-0
Degree $2$
Conductor $1944$
Sign $-0.993 - 0.109i$
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  + (−0.790 + 0.612i)2-s + (−0.627 + 0.778i)3-s + (0.249 − 0.968i)4-s + (0.0193 − 0.999i)6-s + (0.396 + 0.918i)8-s + (−0.211 − 0.977i)9-s + (−0.623 + 1.82i)11-s + (0.597 + 0.802i)12-s + (−0.875 − 0.483i)16-s + (1.03 + 0.245i)17-s + (0.766 + 0.642i)18-s + (−0.902 − 0.956i)19-s + (−0.623 − 1.82i)22-s + (−0.963 − 0.268i)24-s + (0.466 + 0.884i)25-s + ⋯
L(s)  = 1  + (−0.790 + 0.612i)2-s + (−0.627 + 0.778i)3-s + (0.249 − 0.968i)4-s + (0.0193 − 0.999i)6-s + (0.396 + 0.918i)8-s + (−0.211 − 0.977i)9-s + (−0.623 + 1.82i)11-s + (0.597 + 0.802i)12-s + (−0.875 − 0.483i)16-s + (1.03 + 0.245i)17-s + (0.766 + 0.642i)18-s + (−0.902 − 0.956i)19-s + (−0.623 − 1.82i)22-s + (−0.963 − 0.268i)24-s + (0.466 + 0.884i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.993 - 0.109i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.993 - 0.109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4184563713\)
\(L(\frac12)\) \(\approx\) \(0.4184563713\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.790 - 0.612i)T \)
3 \( 1 + (0.627 - 0.778i)T \)
good5 \( 1 + (-0.466 - 0.884i)T^{2} \)
7 \( 1 + (0.360 - 0.932i)T^{2} \)
11 \( 1 + (0.623 - 1.82i)T + (-0.790 - 0.612i)T^{2} \)
13 \( 1 + (0.431 - 0.902i)T^{2} \)
17 \( 1 + (-1.03 - 0.245i)T + (0.893 + 0.448i)T^{2} \)
19 \( 1 + (0.902 + 0.956i)T + (-0.0581 + 0.998i)T^{2} \)
23 \( 1 + (0.627 - 0.778i)T^{2} \)
29 \( 1 + (-0.0968 - 0.995i)T^{2} \)
31 \( 1 + (-0.323 + 0.946i)T^{2} \)
37 \( 1 + (-0.597 + 0.802i)T^{2} \)
41 \( 1 + (0.666 - 0.272i)T + (0.713 - 0.700i)T^{2} \)
43 \( 1 + (1.16 - 1.44i)T + (-0.211 - 0.977i)T^{2} \)
47 \( 1 + (-0.323 - 0.946i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (1.15 - 1.31i)T + (-0.135 - 0.990i)T^{2} \)
61 \( 1 + (0.875 - 0.483i)T^{2} \)
67 \( 1 + (-0.837 + 0.760i)T + (0.0968 - 0.995i)T^{2} \)
71 \( 1 + (0.286 - 0.957i)T^{2} \)
73 \( 1 + (1.89 + 0.221i)T + (0.973 + 0.230i)T^{2} \)
79 \( 1 + (0.963 - 0.268i)T^{2} \)
83 \( 1 + (0.107 + 0.0439i)T + (0.713 + 0.700i)T^{2} \)
89 \( 1 + (0.819 - 1.10i)T + (-0.286 - 0.957i)T^{2} \)
97 \( 1 + (-1.39 + 0.840i)T + (0.466 - 0.884i)T^{2} \)
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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

−9.714530612482056753529396088594, −9.180725524592109712249876270173, −8.204895471751983686961621276654, −7.34793412031717739174820601399, −6.70673404956012960193670907180, −5.82364493197546051869196771556, −4.91637752004260157715086014546, −4.50246762907226166140325643503, −2.93903099209852625581958911596, −1.54612792176039682451449439229, 0.43664253170800265323186546969, 1.68995302673118353719642250265, 2.83169968378225690120829608213, 3.69175756004494035600458467939, 5.11144864071369092261842119673, 5.97749439527051412231003801568, 6.69672990830105286554720973116, 7.67428244382152418015777629304, 8.330939121360847121024596377876, 8.694023008752335999650798564923

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