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Properties

Label 2-1944-72.5-c0-0-4
Degree $2$
Conductor $1944$
Sign $0.996 + 0.0871i$
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  + (−0.258 + 0.965i)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 + (−0.999 − i)10-s + (−0.866 − 0.5i)13-s + (0.965 − 0.258i)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  + (−0.258 + 0.965i)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 + (−0.999 − i)10-s + (−0.866 − 0.5i)13-s + (0.965 − 0.258i)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

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

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.996 + 0.0871i$
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: \((2,\ 1944,\ (\ :0),\ 0.996 + 0.0871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5772512001\)
\(L(\frac12)\) \(\approx\) \(0.5772512001\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
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

−9.486680961326714543496445730008, −8.368823610867092175460920565367, −7.41789217186096111249575733916, −7.10449527943099640521916597309, −6.67791156116936342988014102272, −5.38876251537702016669864655491, −4.62238130718590760422557743047, −3.55859354667974660930774562798, −2.76786616850989997190009254096, −0.51777592074936686656825626001, 1.26585062887823007693960992745, 2.42491946892223110330033464546, 3.52452093217410266029486365937, 4.39530111134870082435361319345, 5.05985208034162147466973579403, 6.03930460299629102976484394198, 7.30143069330940779623978480507, 8.279566167044380690776244576033, 8.680003550558559350596465864076, 9.295087414032468312934219732096

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