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Properties

Label 1-1480-1480.997-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.896 + 0.442i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (0.984 − 0.173i)3-s + (0.642 + 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.939 + 0.342i)13-s + (0.939 − 0.342i)17-s + (−0.984 + 0.173i)19-s + (0.766 + 0.642i)21-s + (−0.5 − 0.866i)23-s + (0.866 − 0.5i)27-s + (0.866 + 0.5i)29-s − i·31-s + (−0.342 + 0.939i)33-s + (0.984 + 0.173i)39-s + (0.939 + 0.342i)41-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)3-s + (0.642 + 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.939 + 0.342i)13-s + (0.939 − 0.342i)17-s + (−0.984 + 0.173i)19-s + (0.766 + 0.642i)21-s + (−0.5 − 0.866i)23-s + (0.866 − 0.5i)27-s + (0.866 + 0.5i)29-s − i·31-s + (−0.342 + 0.939i)33-s + (0.984 + 0.173i)39-s + (0.939 + 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.896 + 0.442i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.896 + 0.442i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.560820136 + 0.5970402781i\)
\(L(\frac12)\) \(\approx\) \(2.560820136 + 0.5970402781i\)
\(L(1)\) \(\approx\) \(1.654403103 + 0.1496897511i\)
\(L(1)\) \(\approx\) \(1.654403103 + 0.1496897511i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 - T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.642 + 0.766i)T \)
59 \( 1 + (-0.642 + 0.766i)T \)
61 \( 1 + (0.342 - 0.939i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.642 - 0.766i)T \)
83 \( 1 + (-0.342 - 0.939i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−20.77214216300178674174535060307, −19.79218333866175096716957197510, −19.28669602103966645633881229797, −18.450046206036716018105761575607, −17.692789278539304869034365221683, −16.766231840557575068049146128893, −15.946702947343016627150761707527, −15.35281723618172320757819534859, −14.36668426515367051872510901044, −13.87998906714487344529685976298, −13.257533706450259496141255328046, −12.39386329740768456838740299864, −11.160045218633715803396871501692, −10.58637529627322329996088025789, −9.9088378352447527312055967030, −8.77009398095330943975232641138, −8.16396356402307127359554671323, −7.69278003376585223977523453171, −6.57732537981852090008132173670, −5.57304015324578234774301307842, −4.55196410368460148352912944086, −3.69790977155262169261702897702, −3.08573652275005701598641119434, −1.87862649459296909555114989785, −0.98916050886957563681300281388, 1.272427919782015046600019475944, 2.160733924756510596941776084804, 2.81013295728292247190136476513, 4.00574296352881125425155405560, 4.681572659631224470469136593150, 5.80283435483283978965608806131, 6.70980628362424792655300334437, 7.71295460331044037724027840044, 8.30786697887108091209733206740, 8.947002434572814568228763244873, 9.85967531279433964250678130496, 10.59882170772887525886160768857, 11.69397284280332468752063612645, 12.48319667046500826979545505040, 13.052866821031127634799329037015, 14.14560185548325691602888345235, 14.54367255320290640292660143856, 15.38286977723146298739985474160, 15.927316174754372191053681168748, 16.9865016265641307302632558705, 18.05858269802272625337310119369, 18.52675471350253332817645295941, 19.035630227741274192015370692394, 20.18633884780523919059383497631, 20.64987570210744045900333864702

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