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Properties

Label 2-992-992.309-c0-0-2
Degree $2$
Conductor $992$
Sign $0.997 + 0.0654i$
Analytic cond. $0.495072$
Root an. cond. $0.703613$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.465 − 1.12i)5-s + (−1.22 − 1.22i)7-s + 0.999i·8-s + (0.707 − 0.707i)9-s + (0.965 − 0.741i)10-s + (−0.448 − 1.67i)14-s + (−0.5 + 0.866i)16-s + (0.965 − 0.258i)18-s + (0.607 + 1.46i)19-s + (1.20 − 0.158i)20-s + (−0.341 − 0.341i)25-s + (0.448 − 1.67i)28-s − 31-s + (−0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.465 − 1.12i)5-s + (−1.22 − 1.22i)7-s + 0.999i·8-s + (0.707 − 0.707i)9-s + (0.965 − 0.741i)10-s + (−0.448 − 1.67i)14-s + (−0.5 + 0.866i)16-s + (0.965 − 0.258i)18-s + (0.607 + 1.46i)19-s + (1.20 − 0.158i)20-s + (−0.341 − 0.341i)25-s + (0.448 − 1.67i)28-s − 31-s + (−0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(992\)    =    \(2^{5} \cdot 31\)
Sign: $0.997 + 0.0654i$
Analytic conductor: \(0.495072\)
Root analytic conductor: \(0.703613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{992} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 992,\ (\ :0),\ 0.997 + 0.0654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.639999910\)
\(L(\frac12)\) \(\approx\) \(1.639999910\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + T \)
good3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.465 + 1.12i)T + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.607 - 1.46i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.758 + 1.83i)T + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + 1.93T + T^{2} \)
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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

−9.887790757838517694700739331488, −9.553135941647053884907234660935, −8.362270951043052201613714995643, −7.43550965150331407937274477343, −6.66417559313472805534074301291, −5.93309342351684281587161859431, −4.92182700729078881893949923378, −3.94776526821846286660685383064, −3.35074072888240447775280518551, −1.42497863346238190275625540084, 2.14784363160746986607182101512, 2.77585656140135736144971730270, 3.69181223763602500258771259411, 5.11673460892500308562119138698, 5.76435826606362302806594842603, 6.82095776407269440175784410227, 7.08557317398827354801451641533, 8.845707665096698485349800301808, 9.678110256335555501743546224988, 10.26803101829061162929038682290

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