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Properties

Label 1-1984-1984.747-r0-0-0
Degree $1$
Conductor $1984$
Sign $0.299 - 0.953i$
Analytic cond. $9.21365$
Root an. cond. $9.21365$
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  + (0.983 − 0.182i)3-s + (−0.991 + 0.130i)5-s + (−0.933 − 0.358i)7-s + (0.933 − 0.358i)9-s + (−0.477 + 0.878i)11-s + (−0.942 − 0.333i)13-s + (−0.951 + 0.309i)15-s + (0.994 − 0.104i)17-s + (−0.430 + 0.902i)19-s + (−0.983 − 0.182i)21-s + (0.156 − 0.987i)23-s + (0.965 − 0.258i)25-s + (0.852 − 0.522i)27-s + (0.0784 + 0.996i)29-s + (−0.309 + 0.951i)33-s + ⋯
L(s)  = 1  + (0.983 − 0.182i)3-s + (−0.991 + 0.130i)5-s + (−0.933 − 0.358i)7-s + (0.933 − 0.358i)9-s + (−0.477 + 0.878i)11-s + (−0.942 − 0.333i)13-s + (−0.951 + 0.309i)15-s + (0.994 − 0.104i)17-s + (−0.430 + 0.902i)19-s + (−0.983 − 0.182i)21-s + (0.156 − 0.987i)23-s + (0.965 − 0.258i)25-s + (0.852 − 0.522i)27-s + (0.0784 + 0.996i)29-s + (−0.309 + 0.951i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.299 - 0.953i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (747, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ 0.299 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.016210273 - 0.7457051826i\)
\(L(\frac12)\) \(\approx\) \(1.016210273 - 0.7457051826i\)
\(L(1)\) \(\approx\) \(1.030522342 - 0.1454124943i\)
\(L(1)\) \(\approx\) \(1.030522342 - 0.1454124943i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.983 - 0.182i)T \)
5 \( 1 + (-0.991 + 0.130i)T \)
7 \( 1 + (-0.933 - 0.358i)T \)
11 \( 1 + (-0.477 + 0.878i)T \)
13 \( 1 + (-0.942 - 0.333i)T \)
17 \( 1 + (0.994 - 0.104i)T \)
19 \( 1 + (-0.430 + 0.902i)T \)
23 \( 1 + (0.156 - 0.987i)T \)
29 \( 1 + (0.0784 + 0.996i)T \)
37 \( 1 + (0.608 + 0.793i)T \)
41 \( 1 + (0.544 - 0.838i)T \)
43 \( 1 + (-0.333 - 0.942i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.688 - 0.725i)T \)
59 \( 1 + (0.566 - 0.824i)T \)
61 \( 1 + (0.923 - 0.382i)T \)
67 \( 1 + (-0.793 - 0.608i)T \)
71 \( 1 + (-0.358 - 0.933i)T \)
73 \( 1 + (-0.629 - 0.777i)T \)
79 \( 1 + (0.994 - 0.104i)T \)
83 \( 1 + (0.566 + 0.824i)T \)
89 \( 1 + (-0.156 - 0.987i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.76996805140606943872572357685, −19.38964268255688626255015580858, −19.10089416158071810910219680758, −18.2156989604970830555430118120, −16.9754518643889222046915895781, −16.15654354392241147601217862775, −15.84024945376762930585763703645, −14.94153778696423378045492122087, −14.498699138559110291506183245038, −13.27612655048450081189169915490, −13.020606366942400770811145114139, −12.01208726304124893141922581504, −11.322265127186791561371843310585, −10.272347465941213352879069657316, −9.53474747212209104045243138019, −8.92874038266280656949206298397, −8.02122360322422360250792627743, −7.54583806899151852529218815907, −6.655473188259064613502813115143, −5.560914672501304244431522047009, −4.5845905554411527361008289437, −3.76668122907742745616167586229, −3.00401268746624365489856477130, −2.48073119889274844979764440919, −0.954951443312090817267846216359, 0.45042591637906997978751271227, 1.83429281893829141748767467653, 2.883514068554029787404247884882, 3.39214017841970289968419282640, 4.26223883198594575308367257817, 5.06271053849868892246412662659, 6.487040384747609380262355528056, 7.1572080281192409500960233883, 7.759352894903477495912877743071, 8.36955353817603176731598614564, 9.432057520702969603702007410190, 10.0992927665801801357639120072, 10.62043102751445164139674496020, 12.08761248255472911599107078734, 12.5155979590543764874739384219, 12.990583709908186417973012511787, 14.160705526588547171037265119819, 14.75433879823264355909610239097, 15.25186053844394645682602813290, 16.18188402379209686302985793920, 16.63426071558506156994085803846, 17.80871962960285067521117706160, 18.727451136395734845338181750560, 19.12239808496063340867641876436, 19.790215771835349395567396217442

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