Properties

Label 2-984-984.11-c0-0-0
Degree $2$
Conductor $984$
Sign $0.871 - 0.489i$
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 + (−0.809 − 0.587i)3-s + (−0.951 + 0.309i)4-s + (−0.453 + 0.891i)6-s + (0.453 + 0.891i)8-s + (0.309 + 0.951i)9-s + (−1.29 + 1.51i)11-s + (0.951 + 0.309i)12-s + (0.809 − 0.587i)16-s + (−0.763 + 0.0600i)17-s + (0.891 − 0.453i)18-s + (−0.891 + 1.45i)19-s + (1.70 + 1.04i)22-s + (0.156 − 0.987i)24-s + (0.587 + 0.809i)25-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (−0.809 − 0.587i)3-s + (−0.951 + 0.309i)4-s + (−0.453 + 0.891i)6-s + (0.453 + 0.891i)8-s + (0.309 + 0.951i)9-s + (−1.29 + 1.51i)11-s + (0.951 + 0.309i)12-s + (0.809 − 0.587i)16-s + (−0.763 + 0.0600i)17-s + (0.891 − 0.453i)18-s + (−0.891 + 1.45i)19-s + (1.70 + 1.04i)22-s + (0.156 − 0.987i)24-s + (0.587 + 0.809i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(984\)    =    \(2^{3} \cdot 3 \cdot 41\)
Sign: $0.871 - 0.489i$
Analytic conductor: \(0.491079\)
Root analytic conductor: \(0.700770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{984} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 984,\ (\ :0),\ 0.871 - 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3785447499\)
\(L(\frac12)\) \(\approx\) \(0.3785447499\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
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

−10.53715093129141361930418066511, −9.772485387378008792172219064998, −8.640386211855204452648993247682, −7.74808841071213171500778733630, −7.07622643579492035592437041022, −5.77521467573370903472350401281, −4.93579885014136031087120387977, −4.12260231977991050374977051599, −2.50661051624844243191407308493, −1.68598565428705395219361286779, 0.41649767305407453135186592383, 2.97578895965750578306977020250, 4.35846217121517345539011792398, 5.00402413004724276959289399618, 5.93613142767764349484780017906, 6.51065185227457844540355019877, 7.53019786709516797344954004118, 8.646697138527176741736235993516, 8.999023319389092169274652674735, 10.25464242602427102269960242811

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