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Properties

Label 2-1944-1944.1627-c0-0-0
Degree $2$
Conductor $1944$
Sign $-0.280 - 0.959i$
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  + (0.952 + 0.305i)2-s + (−0.875 + 0.483i)3-s + (0.813 + 0.581i)4-s + (−0.981 + 0.192i)6-s + (0.597 + 0.802i)8-s + (0.533 − 0.845i)9-s + (−1.79 + 0.281i)11-s + (−0.993 − 0.116i)12-s + (0.323 + 0.946i)16-s + (1.08 + 1.14i)17-s + (0.766 − 0.642i)18-s + (0.360 + 1.20i)19-s + (−1.79 − 0.281i)22-s + (−0.910 − 0.413i)24-s + (−0.135 + 0.990i)25-s + ⋯
L(s)  = 1  + (0.952 + 0.305i)2-s + (−0.875 + 0.483i)3-s + (0.813 + 0.581i)4-s + (−0.981 + 0.192i)6-s + (0.597 + 0.802i)8-s + (0.533 − 0.845i)9-s + (−1.79 + 0.281i)11-s + (−0.993 − 0.116i)12-s + (0.323 + 0.946i)16-s + (1.08 + 1.14i)17-s + (0.766 − 0.642i)18-s + (0.360 + 1.20i)19-s + (−1.79 − 0.281i)22-s + (−0.910 − 0.413i)24-s + (−0.135 + 0.990i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.280 - 0.959i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.280 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.422583392\)
\(L(\frac12)\) \(\approx\) \(1.422583392\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.952 - 0.305i)T \)
3 \( 1 + (0.875 - 0.483i)T \)
good5 \( 1 + (0.135 - 0.990i)T^{2} \)
7 \( 1 + (-0.856 + 0.516i)T^{2} \)
11 \( 1 + (1.79 - 0.281i)T + (0.952 - 0.305i)T^{2} \)
13 \( 1 + (-0.249 + 0.968i)T^{2} \)
17 \( 1 + (-1.08 - 1.14i)T + (-0.0581 + 0.998i)T^{2} \)
19 \( 1 + (-0.360 - 1.20i)T + (-0.835 + 0.549i)T^{2} \)
23 \( 1 + (0.875 - 0.483i)T^{2} \)
29 \( 1 + (0.565 + 0.824i)T^{2} \)
31 \( 1 + (-0.987 + 0.154i)T^{2} \)
37 \( 1 + (0.993 - 0.116i)T^{2} \)
41 \( 1 + (1.26 + 1.15i)T + (0.0968 + 0.995i)T^{2} \)
43 \( 1 + (-1.29 + 0.715i)T + (0.533 - 0.845i)T^{2} \)
47 \( 1 + (-0.987 - 0.154i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (0.406 - 0.503i)T + (-0.211 - 0.977i)T^{2} \)
61 \( 1 + (-0.323 + 0.946i)T^{2} \)
67 \( 1 + (0.897 + 1.70i)T + (-0.565 + 0.824i)T^{2} \)
71 \( 1 + (-0.973 - 0.230i)T^{2} \)
73 \( 1 + (0.791 - 1.83i)T + (-0.686 - 0.727i)T^{2} \)
79 \( 1 + (0.910 - 0.413i)T^{2} \)
83 \( 1 + (-1.23 + 1.12i)T + (0.0968 - 0.995i)T^{2} \)
89 \( 1 + (-0.569 + 0.0665i)T + (0.973 - 0.230i)T^{2} \)
97 \( 1 + (-1.31 + 1.50i)T + (-0.135 - 0.990i)T^{2} \)
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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

−10.04953405820287337420491965472, −8.675142689227311316551752760410, −7.69106525922124150576193765931, −7.27241382873237652750223415379, −5.98687008681610903437969944186, −5.60280976725314395181312558763, −4.97898358443356775851146525372, −3.92034450271573801176054177371, −3.22592657885107830646665242305, −1.81040375633895503690848259427, 0.851709325262539438576015355766, 2.41307532361251350563813796617, 3.04821006451185685083345596605, 4.58038946340431345679358527983, 5.11196094145431604390986164592, 5.74347720464044593986710878808, 6.59542958556029962891671773558, 7.49708160813485383287446854364, 7.907071657648004785349995140162, 9.412514080923194273724261522016

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