Properties

Label 2-984-984.299-c0-0-0
Degree $2$
Conductor $984$
Sign $0.224 - 0.974i$
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 + (−0.809 + 0.587i)3-s + (0.951 + 0.309i)4-s + (0.891 − 0.453i)6-s + (−0.891 − 0.453i)8-s + (0.309 − 0.951i)9-s + (−0.119 + 1.51i)11-s + (−0.951 + 0.309i)12-s + (0.809 + 0.587i)16-s + (0.581 − 0.497i)17-s + (−0.453 + 0.891i)18-s + (0.453 + 0.108i)19-s + (0.355 − 1.47i)22-s + (0.987 − 0.156i)24-s + (−0.587 + 0.809i)25-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)2-s + (−0.809 + 0.587i)3-s + (0.951 + 0.309i)4-s + (0.891 − 0.453i)6-s + (−0.891 − 0.453i)8-s + (0.309 − 0.951i)9-s + (−0.119 + 1.51i)11-s + (−0.951 + 0.309i)12-s + (0.809 + 0.587i)16-s + (0.581 − 0.497i)17-s + (−0.453 + 0.891i)18-s + (0.453 + 0.108i)19-s + (0.355 − 1.47i)22-s + (0.987 − 0.156i)24-s + (−0.587 + 0.809i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(984\)    =    \(2^{3} \cdot 3 \cdot 41\)
Sign: $0.224 - 0.974i$
Analytic conductor: \(0.491079\)
Root analytic conductor: \(0.700770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{984} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 984,\ (\ :0),\ 0.224 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4704073405\)
\(L(\frac12)\) \(\approx\) \(0.4704073405\)
\(L(1)\) 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 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.453 - 0.891i)T \)
good5 \( 1 + (0.587 - 0.809i)T^{2} \)
7 \( 1 + (0.453 - 0.891i)T^{2} \)
11 \( 1 + (0.119 - 1.51i)T + (-0.987 - 0.156i)T^{2} \)
13 \( 1 + (-0.891 + 0.453i)T^{2} \)
17 \( 1 + (-0.581 + 0.497i)T + (0.156 - 0.987i)T^{2} \)
19 \( 1 + (-0.453 - 0.108i)T + (0.891 + 0.453i)T^{2} \)
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 \( 1 + (0.0123 + 0.156i)T + (-0.987 + 0.156i)T^{2} \)
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 \( 1 + (-1.70 + 1.04i)T + (0.453 - 0.891i)T^{2} \)
97 \( 1 + (-1.93 + 0.152i)T + (0.987 - 0.156i)T^{2} \)
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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

−10.08022269104008197745180204152, −9.774077488856989782498240273196, −9.056447848483862789626909681961, −7.75671207214367847399218441155, −7.18436377470570792935872260346, −6.21976057469878928991425979102, −5.24884413129434228880270654084, −4.19754934138758644296517039493, −2.94735650782143123331770181642, −1.43552165517408539887573819544, 0.73450578042483804034159042729, 2.12336391843154232107267260688, 3.49919921949041773282994721990, 5.25441720700913866357364734906, 5.94140736432236681565781767813, 6.62858385556368961513542343756, 7.64505857505098322167505361623, 8.228147565122510719547466167948, 9.078359152559226807493770156316, 10.25766103104345018058726931705

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