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Properties

Label 2-984-984.179-c0-0-0
Degree $2$
Conductor $984$
Sign $0.554 + 0.832i$
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 + (1.29 + 1.51i)11-s + (0.987 + 0.156i)12-s + (0.809 + 0.587i)16-s + (0.763 + 0.0600i)17-s + (−0.156 − 0.987i)18-s + (−0.891 − 1.45i)19-s + (1.70 − 1.04i)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 + (1.29 + 1.51i)11-s + (0.987 + 0.156i)12-s + (0.809 + 0.587i)16-s + (0.763 + 0.0600i)17-s + (−0.156 − 0.987i)18-s + (−0.891 − 1.45i)19-s + (1.70 − 1.04i)22-s + (0.309 − 0.951i)24-s + (0.587 − 0.809i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(984\)    =    \(2^{3} \cdot 3 \cdot 41\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(0.491079\)
Root analytic conductor: \(0.700770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{984} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 984,\ (\ :0),\ 0.554 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7856674231\)
\(L(\frac12)\) \(\approx\) \(0.7856674231\)
\(L(1)\) 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 \( 1 + (-1.29 - 1.51i)T + (-0.156 + 0.987i)T^{2} \)
13 \( 1 + (0.453 + 0.891i)T^{2} \)
17 \( 1 + (-0.763 - 0.0600i)T + (0.987 + 0.156i)T^{2} \)
19 \( 1 + (0.891 + 1.45i)T + (-0.453 + 0.891i)T^{2} \)
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 \( 1 + (0.843 - 0.987i)T + (-0.156 - 0.987i)T^{2} \)
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 \( 1 + (0.355 - 1.47i)T + (-0.891 - 0.453i)T^{2} \)
97 \( 1 + (0.794 + 0.678i)T + (0.156 + 0.987i)T^{2} \)
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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

−10.25403509133705822678716791876, −9.458480175949074480880339944139, −8.859931213900179330269124337239, −7.37663401361559730676146461246, −6.56029129695162513788863754401, −5.56275476445929065862916975460, −4.44412697876590107615237663801, −4.16493786341549593346928867407, −2.49636601451567028148191541970, −1.17885750635250723298046698876, 1.13537652305015730341117688843, 3.51401851994674569706916705819, 4.27324874024378164249199731561, 5.55911743783660018686463531392, 5.98129569787388407297429895727, 6.73001506508943683324170964683, 7.66918661005140623710276860253, 8.529595387200568567878865936077, 9.350104793932005031274092984975, 10.31475271346323334658071966757

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