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Properties

Label 2-1944-1944.475-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  + (0.713 + 0.700i)2-s + (0.952 + 0.305i)3-s + (0.0193 + 0.999i)4-s + (0.466 + 0.884i)6-s + (−0.686 + 0.727i)8-s + (0.813 + 0.581i)9-s + (−1.62 + 0.662i)11-s + (−0.286 + 0.957i)12-s + (−0.999 + 0.0387i)16-s + (1.78 + 0.894i)17-s + (0.173 + 0.984i)18-s + (0.0919 − 1.57i)19-s + (−1.62 − 0.662i)22-s + (−0.875 + 0.483i)24-s + (−0.431 − 0.902i)25-s + ⋯
L(s)  = 1  + (0.713 + 0.700i)2-s + (0.952 + 0.305i)3-s + (0.0193 + 0.999i)4-s + (0.466 + 0.884i)6-s + (−0.686 + 0.727i)8-s + (0.813 + 0.581i)9-s + (−1.62 + 0.662i)11-s + (−0.286 + 0.957i)12-s + (−0.999 + 0.0387i)16-s + (1.78 + 0.894i)17-s + (0.173 + 0.984i)18-s + (0.0919 − 1.57i)19-s + (−1.62 − 0.662i)22-s + (−0.875 + 0.483i)24-s + (−0.431 − 0.902i)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: $\chi_{1944} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.280 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.152047005\)
\(L(\frac12)\) \(\approx\) \(2.152047005\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.713 - 0.700i)T \)
3 \( 1 + (-0.952 - 0.305i)T \)
good5 \( 1 + (0.431 + 0.902i)T^{2} \)
7 \( 1 + (0.211 - 0.977i)T^{2} \)
11 \( 1 + (1.62 - 0.662i)T + (0.713 - 0.700i)T^{2} \)
13 \( 1 + (-0.987 + 0.154i)T^{2} \)
17 \( 1 + (-1.78 - 0.894i)T + (0.597 + 0.802i)T^{2} \)
19 \( 1 + (-0.0919 + 1.57i)T + (-0.993 - 0.116i)T^{2} \)
23 \( 1 + (-0.952 - 0.305i)T^{2} \)
29 \( 1 + (-0.657 - 0.753i)T^{2} \)
31 \( 1 + (-0.925 + 0.378i)T^{2} \)
37 \( 1 + (0.286 + 0.957i)T^{2} \)
41 \( 1 + (-0.407 - 0.113i)T + (0.856 + 0.516i)T^{2} \)
43 \( 1 + (1.83 + 0.588i)T + (0.813 + 0.581i)T^{2} \)
47 \( 1 + (-0.925 - 0.378i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-1.57 + 1.22i)T + (0.249 - 0.968i)T^{2} \)
61 \( 1 + (0.999 + 0.0387i)T^{2} \)
67 \( 1 + (-1.14 + 0.519i)T + (0.657 - 0.753i)T^{2} \)
71 \( 1 + (0.835 + 0.549i)T^{2} \)
73 \( 1 + (-0.188 - 0.0446i)T + (0.893 + 0.448i)T^{2} \)
79 \( 1 + (0.875 + 0.483i)T^{2} \)
83 \( 1 + (-1.91 + 0.532i)T + (0.856 - 0.516i)T^{2} \)
89 \( 1 + (-0.0333 - 0.111i)T + (-0.835 + 0.549i)T^{2} \)
97 \( 1 + (1.04 + 1.66i)T + (-0.431 + 0.902i)T^{2} \)
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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

−9.570160358865756767267329754846, −8.465854221384806540217579477419, −7.986295949303827284101618757528, −7.42460665216808073880607820183, −6.53324164416093780519591330630, −5.31907590309523314962162170502, −4.88656209361074865863780951619, −3.84136934106286308039584195917, −2.99035006045845655241796467279, −2.19801572733610856147082782654, 1.23636005831843819244792140708, 2.41621744961436563907027074358, 3.24600162046332550507056016305, 3.78232703798893073980833874613, 5.23152774366018438211756245290, 5.57932240406920571017343796657, 6.79078981168876856418742424196, 7.81200830373826008881881601042, 8.180878994261568184996030711276, 9.384520166132793710129322564867

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