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Properties

Label 2-1944-1944.1579-c0-0-0
Degree $2$
Conductor $1944$
Sign $-0.505 - 0.862i$
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  + (0.0193 + 0.999i)2-s + (0.813 + 0.581i)3-s + (−0.999 + 0.0387i)4-s + (−0.565 + 0.824i)6-s + (−0.0581 − 0.998i)8-s + (0.323 + 0.946i)9-s + (0.761 − 0.746i)11-s + (−0.835 − 0.549i)12-s + (0.996 − 0.0774i)16-s + (1.17 + 1.58i)17-s + (−0.939 + 0.342i)18-s + (−0.495 − 0.0579i)19-s + (0.761 + 0.746i)22-s + (0.533 − 0.845i)24-s + (−0.627 + 0.778i)25-s + ⋯
L(s)  = 1  + (0.0193 + 0.999i)2-s + (0.813 + 0.581i)3-s + (−0.999 + 0.0387i)4-s + (−0.565 + 0.824i)6-s + (−0.0581 − 0.998i)8-s + (0.323 + 0.946i)9-s + (0.761 − 0.746i)11-s + (−0.835 − 0.549i)12-s + (0.996 − 0.0774i)16-s + (1.17 + 1.58i)17-s + (−0.939 + 0.342i)18-s + (−0.495 − 0.0579i)19-s + (0.761 + 0.746i)22-s + (0.533 − 0.845i)24-s + (−0.627 + 0.778i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.505 - 0.862i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.505 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.435401728\)
\(L(\frac12)\) \(\approx\) \(1.435401728\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0193 - 0.999i)T \)
3 \( 1 + (-0.813 - 0.581i)T \)
good5 \( 1 + (0.627 - 0.778i)T^{2} \)
7 \( 1 + (0.910 + 0.413i)T^{2} \)
11 \( 1 + (-0.761 + 0.746i)T + (0.0193 - 0.999i)T^{2} \)
13 \( 1 + (-0.952 + 0.305i)T^{2} \)
17 \( 1 + (-1.17 - 1.58i)T + (-0.286 + 0.957i)T^{2} \)
19 \( 1 + (0.495 + 0.0579i)T + (0.973 + 0.230i)T^{2} \)
23 \( 1 + (-0.813 - 0.581i)T^{2} \)
29 \( 1 + (0.135 - 0.990i)T^{2} \)
31 \( 1 + (-0.713 + 0.700i)T^{2} \)
37 \( 1 + (0.835 - 0.549i)T^{2} \)
41 \( 1 + (1.55 + 0.940i)T + (0.466 + 0.884i)T^{2} \)
43 \( 1 + (-1.39 - 0.995i)T + (0.323 + 0.946i)T^{2} \)
47 \( 1 + (-0.713 - 0.700i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.497 + 1.93i)T + (-0.875 - 0.483i)T^{2} \)
61 \( 1 + (-0.996 - 0.0774i)T^{2} \)
67 \( 1 + (0.278 - 0.319i)T + (-0.135 - 0.990i)T^{2} \)
71 \( 1 + (-0.396 - 0.918i)T^{2} \)
73 \( 1 + (1.75 + 0.880i)T + (0.597 + 0.802i)T^{2} \)
79 \( 1 + (-0.533 - 0.845i)T^{2} \)
83 \( 1 + (-1.66 + 1.00i)T + (0.466 - 0.884i)T^{2} \)
89 \( 1 + (-1.65 + 1.09i)T + (0.396 - 0.918i)T^{2} \)
97 \( 1 + (0.798 - 1.67i)T + (-0.627 - 0.778i)T^{2} \)
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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

−9.391734365835346128827670302100, −8.723169493666690965715010962389, −8.129908927317125336695221758399, −7.50576393077177110664326825264, −6.41179218278893900210623094577, −5.74613278661825821680297257844, −4.80904963292541476562480362652, −3.76405899790255818807639967538, −3.41815605030981533981258909164, −1.63659857389040203900628548758, 1.10685100496249315764644457687, 2.17243291510251566089913120037, 3.01962746248435907302477106714, 3.92445990126416397955176794777, 4.75048479501500900391360896857, 5.88655395686939803213736926718, 6.97161049384880368365455810900, 7.69433796412813790144539874911, 8.490529620791780501620802877142, 9.287653912583880298191747792991

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