Properties

Label 2-984-984.563-c0-0-1
Degree $2$
Conductor $984$
Sign $0.508 + 0.861i$
Analytic cond. $0.491079$
Root an. cond. $0.700770$
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  + (−0.156 − 0.987i)2-s + (0.987 + 0.156i)3-s + (−0.951 + 0.309i)4-s i·6-s + (0.453 + 0.891i)8-s + (0.951 + 0.309i)9-s + (−0.119 − 0.101i)11-s + (−0.987 + 0.156i)12-s + (0.809 − 0.587i)16-s + (−0.144 − 1.84i)17-s + (0.156 − 0.987i)18-s + (0.891 + 0.546i)19-s + (−0.0819 + 0.133i)22-s + (0.309 + 0.951i)24-s + (0.587 + 0.809i)25-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (0.987 + 0.156i)3-s + (−0.951 + 0.309i)4-s i·6-s + (0.453 + 0.891i)8-s + (0.951 + 0.309i)9-s + (−0.119 − 0.101i)11-s + (−0.987 + 0.156i)12-s + (0.809 − 0.587i)16-s + (−0.144 − 1.84i)17-s + (0.156 − 0.987i)18-s + (0.891 + 0.546i)19-s + (−0.0819 + 0.133i)22-s + (0.309 + 0.951i)24-s + (0.587 + 0.809i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(984\)    =    \(2^{3} \cdot 3 \cdot 41\)
Sign: $0.508 + 0.861i$
Analytic conductor: \(0.491079\)
Root analytic conductor: \(0.700770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{984} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 984,\ (\ :0),\ 0.508 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.213332789\)
\(L(\frac12)\) \(\approx\) \(1.213332789\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.156 + 0.987i)T \)
3 \( 1 + (-0.987 - 0.156i)T \)
41 \( 1 + (0.891 + 0.453i)T \)
good5 \( 1 + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + (0.891 - 0.453i)T^{2} \)
11 \( 1 + (0.119 + 0.101i)T + (0.156 + 0.987i)T^{2} \)
13 \( 1 + (-0.453 + 0.891i)T^{2} \)
17 \( 1 + (0.144 + 1.84i)T + (-0.987 + 0.156i)T^{2} \)
19 \( 1 + (-0.891 - 0.546i)T + (0.453 + 0.891i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.987 + 0.156i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (1.16 - 0.183i)T + (0.951 - 0.309i)T^{2} \)
47 \( 1 + (-0.891 - 0.453i)T^{2} \)
53 \( 1 + (-0.987 - 0.156i)T^{2} \)
59 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.951 + 0.309i)T^{2} \)
67 \( 1 + (1.15 - 0.987i)T + (0.156 - 0.987i)T^{2} \)
71 \( 1 + (0.156 + 0.987i)T^{2} \)
73 \( 1 + (-1.14 + 1.14i)T - iT^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + 0.618iT - T^{2} \)
89 \( 1 + (1.26 - 0.303i)T + (0.891 - 0.453i)T^{2} \)
97 \( 1 + (1.10 + 1.29i)T + (-0.156 + 0.987i)T^{2} \)
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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

−9.886035446453382622064907744658, −9.380438235023222628243937143046, −8.661739444037617210102447797887, −7.76134268114074386140488148993, −7.05000613287504241153657614474, −5.32934544370062284113212298271, −4.57011380967346997556385629507, −3.37195439493531253720698753761, −2.79447339986914763966568913865, −1.47358229936521453418016369015, 1.60382354151829398447131311954, 3.20769466812265692075035799423, 4.18811297717818459472537249880, 5.15394356460842383090198462359, 6.38852154842926891377296702956, 6.95910885779339952301945504832, 8.121058361115834031758767123571, 8.312173006272035974280826024598, 9.334116942720237311547971602592, 9.994104021486125378569130917397

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