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Properties

Label 2-984-984.275-c0-0-0
Degree $2$
Conductor $984$
Sign $-0.385 - 0.922i$
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  + (−0.987 − 0.156i)2-s + (0.156 + 0.987i)3-s + (0.951 + 0.309i)4-s − i·6-s + (−0.891 − 0.453i)8-s + (−0.951 + 0.309i)9-s + (−1.29 − 0.101i)11-s + (−0.156 + 0.987i)12-s + (0.809 + 0.587i)16-s + (1.20 + 1.40i)17-s + (0.987 − 0.156i)18-s + (−0.453 + 1.89i)19-s + (1.26 + 0.303i)22-s + (0.309 − 0.951i)24-s + (−0.587 + 0.809i)25-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)2-s + (0.156 + 0.987i)3-s + (0.951 + 0.309i)4-s − i·6-s + (−0.891 − 0.453i)8-s + (−0.951 + 0.309i)9-s + (−1.29 − 0.101i)11-s + (−0.156 + 0.987i)12-s + (0.809 + 0.587i)16-s + (1.20 + 1.40i)17-s + (0.987 − 0.156i)18-s + (−0.453 + 1.89i)19-s + (1.26 + 0.303i)22-s + (0.309 − 0.951i)24-s + (−0.587 + 0.809i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(984\)    =    \(2^{3} \cdot 3 \cdot 41\)
Sign: $-0.385 - 0.922i$
Analytic conductor: \(0.491079\)
Root analytic conductor: \(0.700770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{984} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 984,\ (\ :0),\ -0.385 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5515709021\)
\(L(\frac12)\) \(\approx\) \(0.5515709021\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.987 + 0.156i)T \)
3 \( 1 + (-0.156 - 0.987i)T \)
41 \( 1 + (-0.453 - 0.891i)T \)
good5 \( 1 + (0.587 - 0.809i)T^{2} \)
7 \( 1 + (-0.453 + 0.891i)T^{2} \)
11 \( 1 + (1.29 + 0.101i)T + (0.987 + 0.156i)T^{2} \)
13 \( 1 + (0.891 - 0.453i)T^{2} \)
17 \( 1 + (-1.20 - 1.40i)T + (-0.156 + 0.987i)T^{2} \)
19 \( 1 + (0.453 - 1.89i)T + (-0.891 - 0.453i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.156 + 0.987i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.183 + 1.16i)T + (-0.951 - 0.309i)T^{2} \)
47 \( 1 + (0.453 + 0.891i)T^{2} \)
53 \( 1 + (-0.156 - 0.987i)T^{2} \)
59 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.951 + 0.309i)T^{2} \)
67 \( 1 + (1.98 - 0.156i)T + (0.987 - 0.156i)T^{2} \)
71 \( 1 + (0.987 + 0.156i)T^{2} \)
73 \( 1 + (-1.14 + 1.14i)T - iT^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + 0.618iT - T^{2} \)
89 \( 1 + (-0.0819 - 0.133i)T + (-0.453 + 0.891i)T^{2} \)
97 \( 1 + (0.0366 + 0.465i)T + (-0.987 + 0.156i)T^{2} \)
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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

−10.31073283010945651172287585952, −9.884298554219761759885913538800, −8.791067181542737566296968711843, −8.086087239507333700600748570479, −7.59902156198885588727602503301, −5.99346682112592329854553330386, −5.53439912318798759467292032028, −3.95391695189209167574991302925, −3.18505980882918029433085515407, −1.88738505720139685521230975679, 0.68031397855853713896612268016, 2.35073133612271114269729541947, 2.93155919767655518622064597354, 5.00378348533960434786210153324, 5.88382683180957757774622016501, 6.89861871060567017705903882077, 7.52418678000988443355885276191, 8.115808323047208071032618313927, 9.035760454622532234946022340876, 9.758128742567770868334299899155

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