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Properties

Label 1-1984-1984.1315-r0-0-0
Degree $1$
Conductor $1984$
Sign $0.345 - 0.938i$
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  + (0.902 − 0.430i)3-s + (0.991 + 0.130i)5-s + (−0.629 − 0.777i)7-s + (0.629 − 0.777i)9-s + (−0.688 + 0.725i)11-s + (−0.566 + 0.824i)13-s + (0.951 − 0.309i)15-s + (0.406 − 0.913i)17-s + (0.182 − 0.983i)19-s + (−0.902 − 0.430i)21-s + (0.987 + 0.156i)23-s + (0.965 + 0.258i)25-s + (0.233 − 0.972i)27-s + (0.649 − 0.760i)29-s + (−0.309 + 0.951i)33-s + ⋯
L(s)  = 1  + (0.902 − 0.430i)3-s + (0.991 + 0.130i)5-s + (−0.629 − 0.777i)7-s + (0.629 − 0.777i)9-s + (−0.688 + 0.725i)11-s + (−0.566 + 0.824i)13-s + (0.951 − 0.309i)15-s + (0.406 − 0.913i)17-s + (0.182 − 0.983i)19-s + (−0.902 − 0.430i)21-s + (0.987 + 0.156i)23-s + (0.965 + 0.258i)25-s + (0.233 − 0.972i)27-s + (0.649 − 0.760i)29-s + (−0.309 + 0.951i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.345 - 0.938i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (1315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ 0.345 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.033491213 - 1.417493802i\)
\(L(\frac12)\) \(\approx\) \(2.033491213 - 1.417493802i\)
\(L(1)\) \(\approx\) \(1.509282428 - 0.4287459277i\)
\(L(1)\) \(\approx\) \(1.509282428 - 0.4287459277i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.902 - 0.430i)T \)
5 \( 1 + (0.991 + 0.130i)T \)
7 \( 1 + (-0.629 - 0.777i)T \)
11 \( 1 + (-0.688 + 0.725i)T \)
13 \( 1 + (-0.566 + 0.824i)T \)
17 \( 1 + (0.406 - 0.913i)T \)
19 \( 1 + (0.182 - 0.983i)T \)
23 \( 1 + (0.987 + 0.156i)T \)
29 \( 1 + (0.649 - 0.760i)T \)
37 \( 1 + (-0.608 + 0.793i)T \)
41 \( 1 + (0.0523 - 0.998i)T \)
43 \( 1 + (-0.824 + 0.566i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (-0.477 - 0.878i)T \)
59 \( 1 + (0.942 + 0.333i)T \)
61 \( 1 + (-0.923 - 0.382i)T \)
67 \( 1 + (0.793 - 0.608i)T \)
71 \( 1 + (0.777 + 0.629i)T \)
73 \( 1 + (-0.933 - 0.358i)T \)
79 \( 1 + (0.406 - 0.913i)T \)
83 \( 1 + (0.942 - 0.333i)T \)
89 \( 1 + (-0.987 + 0.156i)T \)
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

−20.17714692922771384108476675088, −19.34807299656100115934537441020, −18.736756262868708102885838004535, −18.12529378284127305289407678174, −17.05250917675199411449034321048, −16.424508424659191316754644349, −15.65793739070033747885068897660, −14.94831525652432151633792357437, −14.30187752490493933260153746161, −13.5314253239008415395655852048, −12.75260080348558932625638761539, −12.42641022173571381378948445633, −10.85622857783077282542708381632, −10.25341568083312335035090164860, −9.72802404576520777276451792988, −8.818473737144363484557357883968, −8.37436148456925691210935127978, −7.43175656400494575504691848920, −6.304942667187600673166350637318, −5.51196305532321032381770367963, −5.00819777995848755631473465586, −3.60905464895003314513810599789, −2.944674327353706749723598951860, −2.33729841571080523034668308778, −1.25711551070225886829985119254, 0.762383948793151093935016815323, 1.876267292169908207290147117070, 2.6353837145026309956868987307, 3.26129845729706183380969221915, 4.4970578660479862645133092046, 5.17572477025922461748612741396, 6.5306700709903370546338846040, 6.99013892515525374549264803193, 7.54414148283939241334148317010, 8.71442456561022107047594616579, 9.57325636574967113742364256853, 9.80105982001343502717446026667, 10.69763685383392496285007760164, 11.87999809298489406684504141911, 12.70075042168759621801279312182, 13.4967022766449650764553169392, 13.70838803222003452036359946743, 14.54598122048364681268549949416, 15.339078896712484760643627676940, 16.13509036145009606656974774318, 17.14122262512504052307944429270, 17.60431039200437580470565722680, 18.51977145894582017939197459374, 19.08041392217042336043922386591, 19.8419811006627090767623037500

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