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Properties

Label 2-1984-1984.1549-c0-0-1
Degree $2$
Conductor $1984$
Sign $0.634 + 0.773i$
Analytic cond. $0.990144$
Root an. cond. $0.995060$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.630 − 0.125i)5-s + (−0.198 − 0.478i)7-s + (−0.707 + 0.707i)8-s + (0.382 − 0.923i)9-s + (−0.576 + 0.284i)10-s + (0.315 + 0.410i)14-s + (0.500 − 0.866i)16-s + (−0.130 + 0.991i)18-s + (0.172 − 0.867i)19-s + (0.483 − 0.423i)20-s + (−0.542 + 0.224i)25-s + (−0.410 − 0.315i)28-s − i·31-s + (−0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.630 − 0.125i)5-s + (−0.198 − 0.478i)7-s + (−0.707 + 0.707i)8-s + (0.382 − 0.923i)9-s + (−0.576 + 0.284i)10-s + (0.315 + 0.410i)14-s + (0.500 − 0.866i)16-s + (−0.130 + 0.991i)18-s + (0.172 − 0.867i)19-s + (0.483 − 0.423i)20-s + (−0.542 + 0.224i)25-s + (−0.410 − 0.315i)28-s − i·31-s + (−0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.634 + 0.773i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1984,\ (\ :0),\ 0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8143422367\)
\(L(\frac12)\) \(\approx\) \(0.8143422367\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + iT \)
good3 \( 1 + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (-0.630 + 0.125i)T + (0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.198 + 0.478i)T + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (-0.923 - 0.382i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.172 + 0.867i)T + (-0.923 - 0.382i)T^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (-1.78 - 0.739i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.382 - 0.923i)T^{2} \)
47 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
53 \( 1 + (0.382 + 0.923i)T^{2} \)
59 \( 1 + (1.47 - 0.293i)T + (0.923 - 0.382i)T^{2} \)
61 \( 1 + (-0.382 + 0.923i)T^{2} \)
67 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
71 \( 1 + (-0.465 - 1.12i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (0.923 + 0.382i)T^{2} \)
89 \( 1 + (-0.707 + 0.707i)T^{2} \)
97 \( 1 + 0.261iT - T^{2} \)
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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

−9.502474561607272384688808804617, −8.593566237957209218884472050319, −7.65075064387128814292165713838, −6.97007698165979099928725611318, −6.26365933877849939430702983824, −5.56298233733441990194052361037, −4.36097002739077004600581928556, −3.18951898906879175009736248541, −2.04022298346660906079901100631, −0.839356523966639805400924372800, 1.54438261392630210012050166296, 2.34185296333308841282916636871, 3.31853239161650625371301037613, 4.58674636958652175222886374130, 5.77269575606629471583694133484, 6.30853663767889141459617832075, 7.41900377387888002692045610247, 7.903154435219559341339425159493, 8.821177743565589317264727447821, 9.508294809713880603485331038382

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