Properties

Label 2-984-984.971-c0-0-0
Degree $2$
Conductor $984$
Sign $-0.849 - 0.526i$
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  + (−0.453 + 0.891i)2-s + (0.309 + 0.951i)3-s + (−0.587 − 0.809i)4-s + (−0.987 − 0.156i)6-s + (0.987 − 0.156i)8-s + (−0.809 + 0.587i)9-s + (0.243 + 0.398i)11-s + (0.587 − 0.809i)12-s + (−0.309 + 0.951i)16-s + (−0.178 + 0.744i)17-s + (−0.156 − 0.987i)18-s + (0.156 + 1.98i)19-s + (−0.465 + 0.0366i)22-s + (0.453 + 0.891i)24-s + (−0.951 − 0.309i)25-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)2-s + (0.309 + 0.951i)3-s + (−0.587 − 0.809i)4-s + (−0.987 − 0.156i)6-s + (0.987 − 0.156i)8-s + (−0.809 + 0.587i)9-s + (0.243 + 0.398i)11-s + (0.587 − 0.809i)12-s + (−0.309 + 0.951i)16-s + (−0.178 + 0.744i)17-s + (−0.156 − 0.987i)18-s + (0.156 + 1.98i)19-s + (−0.465 + 0.0366i)22-s + (0.453 + 0.891i)24-s + (−0.951 − 0.309i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(984\)    =    \(2^{3} \cdot 3 \cdot 41\)
Sign: $-0.849 - 0.526i$
Analytic conductor: \(0.491079\)
Root analytic conductor: \(0.700770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{984} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 984,\ (\ :0),\ -0.849 - 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7813324669\)
\(L(\frac12)\) \(\approx\) \(0.7813324669\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 - 0.891i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.156 + 0.987i)T \)
good5 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 + (0.156 + 0.987i)T^{2} \)
11 \( 1 + (-0.243 - 0.398i)T + (-0.453 + 0.891i)T^{2} \)
13 \( 1 + (0.987 + 0.156i)T^{2} \)
17 \( 1 + (0.178 - 0.744i)T + (-0.891 - 0.453i)T^{2} \)
19 \( 1 + (-0.156 - 1.98i)T + (-0.987 + 0.156i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.891 - 0.453i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-1.69 - 0.863i)T + (0.587 + 0.809i)T^{2} \)
47 \( 1 + (-0.156 + 0.987i)T^{2} \)
53 \( 1 + (-0.891 + 0.453i)T^{2} \)
59 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.587 - 0.809i)T^{2} \)
67 \( 1 + (0.546 - 0.891i)T + (-0.453 - 0.891i)T^{2} \)
71 \( 1 + (-0.453 + 0.891i)T^{2} \)
73 \( 1 + (-0.437 + 0.437i)T - iT^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 + 1.61iT - T^{2} \)
89 \( 1 + (1.29 + 1.10i)T + (0.156 + 0.987i)T^{2} \)
97 \( 1 + (0.133 + 0.0819i)T + (0.453 + 0.891i)T^{2} \)
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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

−10.13927027346640801983080172359, −9.764554749086130906179758744987, −8.798013287826193926385408075965, −8.144135853956800977112706271174, −7.39565770213875950803598976331, −6.10631349747722771828369458715, −5.57053822079153323678944839653, −4.37378185796687587781450717186, −3.72505838923294495172294020406, −1.92673301335269606055719707329, 0.873926974778355221834369820291, 2.30663092180870447473393523714, 3.06102894412905703312115605950, 4.28040025314901041648342010859, 5.50775899219876150600973274158, 6.79126532027886940585537525623, 7.44985024576350242121841741602, 8.324156865466653980407701407971, 9.139133262131429339809497411420, 9.592491684510101583554305059589

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