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Properties

Label 2-1944-1944.1267-c0-0-0
Degree $2$
Conductor $1944$
Sign $0.761 + 0.647i$
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  + (−0.981 − 0.192i)2-s + (0.996 + 0.0774i)3-s + (0.925 + 0.378i)4-s + (−0.963 − 0.268i)6-s + (−0.835 − 0.549i)8-s + (0.987 + 0.154i)9-s + (−0.153 − 1.57i)11-s + (0.893 + 0.448i)12-s + (0.713 + 0.700i)16-s + (−0.0385 − 0.00450i)17-s + (−0.939 − 0.342i)18-s + (0.644 − 1.49i)19-s + (−0.153 + 1.57i)22-s + (−0.790 − 0.612i)24-s + (−0.875 + 0.483i)25-s + ⋯
L(s)  = 1  + (−0.981 − 0.192i)2-s + (0.996 + 0.0774i)3-s + (0.925 + 0.378i)4-s + (−0.963 − 0.268i)6-s + (−0.835 − 0.549i)8-s + (0.987 + 0.154i)9-s + (−0.153 − 1.57i)11-s + (0.893 + 0.448i)12-s + (0.713 + 0.700i)16-s + (−0.0385 − 0.00450i)17-s + (−0.939 − 0.342i)18-s + (0.644 − 1.49i)19-s + (−0.153 + 1.57i)22-s + (−0.790 − 0.612i)24-s + (−0.875 + 0.483i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.761 + 0.647i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.761 + 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.081306496\)
\(L(\frac12)\) \(\approx\) \(1.081306496\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.981 + 0.192i)T \)
3 \( 1 + (-0.996 - 0.0774i)T \)
good5 \( 1 + (0.875 - 0.483i)T^{2} \)
7 \( 1 + (0.431 - 0.902i)T^{2} \)
11 \( 1 + (0.153 + 1.57i)T + (-0.981 + 0.192i)T^{2} \)
13 \( 1 + (0.999 - 0.0387i)T^{2} \)
17 \( 1 + (0.0385 + 0.00450i)T + (0.973 + 0.230i)T^{2} \)
19 \( 1 + (-0.644 + 1.49i)T + (-0.686 - 0.727i)T^{2} \)
23 \( 1 + (-0.996 - 0.0774i)T^{2} \)
29 \( 1 + (0.211 - 0.977i)T^{2} \)
31 \( 1 + (-0.0968 - 0.995i)T^{2} \)
37 \( 1 + (-0.893 + 0.448i)T^{2} \)
41 \( 1 + (0.567 + 0.650i)T + (-0.135 + 0.990i)T^{2} \)
43 \( 1 + (-1.31 - 0.101i)T + (0.987 + 0.154i)T^{2} \)
47 \( 1 + (-0.0968 + 0.995i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-1.16 - 0.830i)T + (0.323 + 0.946i)T^{2} \)
61 \( 1 + (-0.713 + 0.700i)T^{2} \)
67 \( 1 + (0.669 - 0.830i)T + (-0.211 - 0.977i)T^{2} \)
71 \( 1 + (-0.597 - 0.802i)T^{2} \)
73 \( 1 + (-0.0418 + 0.719i)T + (-0.993 - 0.116i)T^{2} \)
79 \( 1 + (0.790 - 0.612i)T^{2} \)
83 \( 1 + (0.902 - 1.03i)T + (-0.135 - 0.990i)T^{2} \)
89 \( 1 + (-0.707 + 0.355i)T + (0.597 - 0.802i)T^{2} \)
97 \( 1 + (0.369 - 1.43i)T + (-0.875 - 0.483i)T^{2} \)
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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

−9.057541406051124958005503803845, −8.754051803137449635724477648909, −7.85504938395608026168134784124, −7.31179180891345878644330044080, −6.38213657863692390890626529528, −5.38244706339754226206056694458, −3.97418449038473842239826249538, −3.12047617287700574451209617386, −2.43079174250397745834215946509, −1.05147743800247709565359622250, 1.58100881133525725511950425197, 2.27241834386899871854087320352, 3.42569275955749414491161076947, 4.47716793926376580228781492576, 5.65224502000472091979972408290, 6.68199403801662375747840885857, 7.41194432347054727856765578871, 7.924539128066640677282776327369, 8.602446918607814679496818344549, 9.658931322141996418252212589817

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