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Properties

Label 2-1944-1944.1555-c0-0-0
Degree $2$
Conductor $1944$
Sign $0.971 + 0.236i$
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  + (−0.431 − 0.902i)2-s + (−0.135 + 0.990i)3-s + (−0.627 + 0.778i)4-s + (0.952 − 0.305i)6-s + (0.973 + 0.230i)8-s + (−0.963 − 0.268i)9-s + (−0.384 − 0.609i)11-s + (−0.686 − 0.727i)12-s + (−0.211 − 0.977i)16-s + (1.52 − 1.00i)17-s + (0.173 + 0.984i)18-s + (0.832 − 0.418i)19-s + (−0.384 + 0.609i)22-s + (−0.360 + 0.932i)24-s + (0.0968 + 0.995i)25-s + ⋯
L(s)  = 1  + (−0.431 − 0.902i)2-s + (−0.135 + 0.990i)3-s + (−0.627 + 0.778i)4-s + (0.952 − 0.305i)6-s + (0.973 + 0.230i)8-s + (−0.963 − 0.268i)9-s + (−0.384 − 0.609i)11-s + (−0.686 − 0.727i)12-s + (−0.211 − 0.977i)16-s + (1.52 − 1.00i)17-s + (0.173 + 0.984i)18-s + (0.832 − 0.418i)19-s + (−0.384 + 0.609i)22-s + (−0.360 + 0.932i)24-s + (0.0968 + 0.995i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.971 + 0.236i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1555, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.971 + 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8279211913\)
\(L(\frac12)\) \(\approx\) \(0.8279211913\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.431 + 0.902i)T \)
3 \( 1 + (0.135 - 0.990i)T \)
good5 \( 1 + (-0.0968 - 0.995i)T^{2} \)
7 \( 1 + (-0.925 - 0.378i)T^{2} \)
11 \( 1 + (0.384 + 0.609i)T + (-0.431 + 0.902i)T^{2} \)
13 \( 1 + (-0.657 + 0.753i)T^{2} \)
17 \( 1 + (-1.52 + 1.00i)T + (0.396 - 0.918i)T^{2} \)
19 \( 1 + (-0.832 + 0.418i)T + (0.597 - 0.802i)T^{2} \)
23 \( 1 + (0.135 - 0.990i)T^{2} \)
29 \( 1 + (-0.0193 - 0.999i)T^{2} \)
31 \( 1 + (-0.533 - 0.845i)T^{2} \)
37 \( 1 + (0.686 - 0.727i)T^{2} \)
41 \( 1 + (-1.84 + 0.143i)T + (0.987 - 0.154i)T^{2} \)
43 \( 1 + (0.269 - 1.97i)T + (-0.963 - 0.268i)T^{2} \)
47 \( 1 + (-0.533 + 0.845i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.197 + 0.374i)T + (-0.565 + 0.824i)T^{2} \)
61 \( 1 + (0.211 - 0.977i)T^{2} \)
67 \( 1 + (1.40 - 1.37i)T + (0.0193 - 0.999i)T^{2} \)
71 \( 1 + (0.0581 - 0.998i)T^{2} \)
73 \( 1 + (0.143 + 0.477i)T + (-0.835 + 0.549i)T^{2} \)
79 \( 1 + (0.360 + 0.932i)T^{2} \)
83 \( 1 + (-1.19 - 0.0925i)T + (0.987 + 0.154i)T^{2} \)
89 \( 1 + (1.22 - 1.30i)T + (-0.0581 - 0.998i)T^{2} \)
97 \( 1 + (-1.29 + 1.17i)T + (0.0968 - 0.995i)T^{2} \)
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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

−9.491248786764050560752309456719, −8.940408920005138082988427051364, −7.942138851625854401791795395502, −7.35953711020515559868862614910, −5.82171751435344741825616587802, −5.16991930987292242390906957584, −4.30194521583416121466354152716, −3.21620973842251362338738063571, −2.83603245194574785664682227328, −1.00026187540506656037902558276, 1.03392599502937352250913684613, 2.18147089052821655427179530529, 3.63813297580976705910331008015, 4.89853823666786903779028707495, 5.75292055918613018202217981618, 6.21201611380505731265090855861, 7.36817601873540528706164368075, 7.60043153966528403285691153867, 8.389217985480905619167401136235, 9.163468764033312380014896126094

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