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Properties

Label 1-1480-1480.243-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.659 + 0.751i$
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  + (0.642 + 0.766i)3-s + (0.342 − 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.984 + 0.173i)17-s + (0.766 − 0.642i)19-s + (0.939 − 0.342i)21-s + (0.866 − 0.5i)23-s + (−0.866 + 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (−0.984 + 0.173i)33-s + (−0.766 − 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + (0.342 − 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.984 + 0.173i)17-s + (0.766 − 0.642i)19-s + (0.939 − 0.342i)21-s + (0.866 − 0.5i)23-s + (−0.866 + 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (−0.984 + 0.173i)33-s + (−0.766 − 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.659 + 0.751i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.659 + 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.857082329 + 0.8415689600i\)
\(L(\frac12)\) \(\approx\) \(1.857082329 + 0.8415689600i\)
\(L(1)\) \(\approx\) \(1.327314154 + 0.3255981807i\)
\(L(1)\) \(\approx\) \(1.327314154 + 0.3255981807i\)

Euler product

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

−20.70366669509374704293520941029, −19.52776796406691151029442171944, −19.109267834446247421671341128826, −18.411402957031684601065970782165, −17.8018849020926897472447680437, −16.88846898642128514350444249141, −15.91485246302022076110599878012, −15.1594837343857176902257997913, −14.375172408702224946240118227812, −13.88753621727245479422674970574, −12.881050010489102059325907596996, −12.19702781860201499220227906764, −11.69541514275130667563362534699, −10.56270430999525852108186060677, −9.52797429478460112345935833458, −8.91136587618460672773245639608, −7.926790148349768088068906998179, −7.62520406382437311795046693181, −6.44991143978970728321608173687, −5.589866940736753883269307687990, −4.91245132138698005964518627206, −3.28670320582666240021147098712, −2.93220175373309687551326939275, −1.8882904616351737550709998025, −0.86251141844480873607581867585, 1.03818460869282135890793085814, 2.37409389775443694207261109782, 3.02721001865100221744664703090, 4.22790260030533324380646775487, 4.686069869894695717972367588596, 5.505517229984708773522811484, 7.02940578230634381734413082715, 7.52688478771359769770850377226, 8.30341058202320715176162789437, 9.3477413450917613294129366394, 10.08149798922367879092353244821, 10.44626395647787652993235510204, 11.52081269233238104948461339783, 12.3836404160623962993436505861, 13.44235707290126855261783654568, 13.96943369405460676754427941920, 14.8413523280749508394020033395, 15.28477132279235740414201451045, 16.25442159685274264258930016195, 17.011830817027936027867941195296, 17.520958627167479722767909495234, 18.66049460583597870345434931519, 19.48168822294350087028521151981, 20.110975417122576583094195919901, 20.75765584941445935909077795977

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