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Properties

Label 2-1944-1944.787-c0-0-0
Degree $2$
Conductor $1944$
Sign $-0.690 - 0.722i$
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  + (−0.740 − 0.672i)2-s + (0.0193 + 0.999i)3-s + (0.0968 + 0.995i)4-s + (0.657 − 0.753i)6-s + (0.597 − 0.802i)8-s + (−0.999 + 0.0387i)9-s + (−0.586 − 1.51i)11-s + (−0.993 + 0.116i)12-s + (−0.981 + 0.192i)16-s + (−1.27 + 1.34i)17-s + (0.766 + 0.642i)18-s + (−0.566 + 1.89i)19-s + (−0.586 + 1.51i)22-s + (0.813 + 0.581i)24-s + (−0.790 + 0.612i)25-s + ⋯
L(s)  = 1  + (−0.740 − 0.672i)2-s + (0.0193 + 0.999i)3-s + (0.0968 + 0.995i)4-s + (0.657 − 0.753i)6-s + (0.597 − 0.802i)8-s + (−0.999 + 0.0387i)9-s + (−0.586 − 1.51i)11-s + (−0.993 + 0.116i)12-s + (−0.981 + 0.192i)16-s + (−1.27 + 1.34i)17-s + (0.766 + 0.642i)18-s + (−0.566 + 1.89i)19-s + (−0.586 + 1.51i)22-s + (0.813 + 0.581i)24-s + (−0.790 + 0.612i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.690 - 0.722i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.690 - 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3531720440\)
\(L(\frac12)\) \(\approx\) \(0.3531720440\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.740 + 0.672i)T \)
3 \( 1 + (-0.0193 - 0.999i)T \)
good5 \( 1 + (0.790 - 0.612i)T^{2} \)
7 \( 1 + (0.875 - 0.483i)T^{2} \)
11 \( 1 + (0.586 + 1.51i)T + (-0.740 + 0.672i)T^{2} \)
13 \( 1 + (-0.713 + 0.700i)T^{2} \)
17 \( 1 + (1.27 - 1.34i)T + (-0.0581 - 0.998i)T^{2} \)
19 \( 1 + (0.566 - 1.89i)T + (-0.835 - 0.549i)T^{2} \)
23 \( 1 + (-0.0193 - 0.999i)T^{2} \)
29 \( 1 + (0.431 + 0.902i)T^{2} \)
31 \( 1 + (0.360 + 0.932i)T^{2} \)
37 \( 1 + (0.993 + 0.116i)T^{2} \)
41 \( 1 + (-0.370 - 1.71i)T + (-0.910 + 0.413i)T^{2} \)
43 \( 1 + (0.00821 + 0.423i)T + (-0.999 + 0.0387i)T^{2} \)
47 \( 1 + (0.360 - 0.932i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (1.93 - 0.303i)T + (0.952 - 0.305i)T^{2} \)
61 \( 1 + (0.981 + 0.192i)T^{2} \)
67 \( 1 + (-0.266 - 0.422i)T + (-0.431 + 0.902i)T^{2} \)
71 \( 1 + (-0.973 + 0.230i)T^{2} \)
73 \( 1 + (-0.369 - 0.855i)T + (-0.686 + 0.727i)T^{2} \)
79 \( 1 + (-0.813 + 0.581i)T^{2} \)
83 \( 1 + (-0.353 + 1.63i)T + (-0.910 - 0.413i)T^{2} \)
89 \( 1 + (-0.569 - 0.0665i)T + (0.973 + 0.230i)T^{2} \)
97 \( 1 + (0.366 - 1.07i)T + (-0.790 - 0.612i)T^{2} \)
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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

−9.746133683448575561202435291582, −8.934113858423574259947254195905, −8.245756313179839252622336480789, −7.921172278497309800852465001527, −6.31933677443163089850556031987, −5.79578728604964322910805986754, −4.45055903288958272282818585700, −3.72824203384107851763257176742, −3.00117332694367254953610068312, −1.76294284926442485767737633055, 0.30313907620589986566798070849, 2.03327589534194710564902325558, 2.53915308208908182908355199965, 4.59453573604096616656553308542, 5.11375761587991048015918259327, 6.32619759732378440456749188807, 6.93042909833610867055407714791, 7.39189538990942503105200032365, 8.152576013385969163225901299588, 9.119805144248102530487732619370

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