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Properties

Label 2-1944-72.5-c0-0-6
Degree $2$
Conductor $1944$
Sign $-0.0871 + 0.996i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 − 1.22i)5-s + (−0.5 − 0.866i)7-s + (−0.707 − 0.707i)8-s + (−1 + 0.999i)10-s + (0.866 + 0.5i)13-s + (0.258 + 0.965i)14-s + (0.500 + 0.866i)16-s − 1.41i·17-s + i·19-s + (1.22 − 0.707i)20-s + (1.22 + 0.707i)23-s + (−0.499 − 0.866i)25-s + (−0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 − 1.22i)5-s + (−0.5 − 0.866i)7-s + (−0.707 − 0.707i)8-s + (−1 + 0.999i)10-s + (0.866 + 0.5i)13-s + (0.258 + 0.965i)14-s + (0.500 + 0.866i)16-s − 1.41i·17-s + i·19-s + (1.22 − 0.707i)20-s + (1.22 + 0.707i)23-s + (−0.499 − 0.866i)25-s + (−0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.0871 + 0.996i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.0871 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8249969320\)
\(L(\frac12)\) \(\approx\) \(0.8249969320\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
good5 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.157085285316082787611720480688, −8.708418215678493575065104935232, −7.69244326551023069338123919204, −7.00075546215871534316893416724, −6.12785441529479088623986569268, −5.22839714753732176487143586336, −4.10138376469841561499873327468, −3.14676131367927703229907701629, −1.76723085765776005991735677568, −0.882020100585218365574229804292, 1.58271735753355434006350704839, 2.74075989518355010926395928147, 3.22985522310687576205902997524, 5.07670138244647809265358177056, 6.04051607185720938837610339893, 6.50337842887078515925058481659, 7.01905870967513726921706910364, 8.360068078949445030810122285363, 8.647538274679332240369219955481, 9.639229296728215519822864626537

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