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Properties

Label 2-984-984.971-c0-0-1
Degree $2$
Conductor $984$
Sign $-0.387 + 0.921i$
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 + (0.891 − 0.453i)3-s + (−0.587 − 0.809i)4-s − 1.00i·6-s + (−0.987 + 0.156i)8-s + (0.587 − 0.809i)9-s + (−0.243 − 0.398i)11-s + (−0.891 − 0.453i)12-s + (−0.309 + 0.951i)16-s + (0.178 − 0.744i)17-s + (−0.453 − 0.891i)18-s + (0.156 + 1.98i)19-s + (−0.465 + 0.0366i)22-s + (−0.809 + 0.587i)24-s + (−0.951 − 0.309i)25-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (0.891 − 0.453i)3-s + (−0.587 − 0.809i)4-s − 1.00i·6-s + (−0.987 + 0.156i)8-s + (0.587 − 0.809i)9-s + (−0.243 − 0.398i)11-s + (−0.891 − 0.453i)12-s + (−0.309 + 0.951i)16-s + (0.178 − 0.744i)17-s + (−0.453 − 0.891i)18-s + (0.156 + 1.98i)19-s + (−0.465 + 0.0366i)22-s + (−0.809 + 0.587i)24-s + (−0.951 − 0.309i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(984\)    =    \(2^{3} \cdot 3 \cdot 41\)
Sign: $-0.387 + 0.921i$
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: \((2,\ 984,\ (\ :0),\ -0.387 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.532281943\)
\(L(\frac12)\) \(\approx\) \(1.532281943\)
\(L(1)\) 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 \( 1 + (-0.891 + 0.453i)T \)
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

−9.865719105109167265335969086188, −9.347490215049721861329266945224, −8.285901743964451684094096745808, −7.68361049882649964485349088417, −6.36548522210061036954231822793, −5.57546601729552557208531084997, −4.27579859835997234931873840612, −3.44986082703839881642989797350, −2.52579859214845822385762480440, −1.37405933941348794764124086923, 2.35375078997196806621935713207, 3.43183578968790687640832923767, 4.36833280667745079212690389269, 5.11548925391970684433381381305, 6.20565507553799522415306037059, 7.34659060724978533538584876599, 7.73387293614616826893097299351, 8.900039997861654439397831588250, 9.202866360343592621511799546047, 10.26626573860876003418804490013

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