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Properties

Label 1-2240-2240.19-r0-0-0
Degree $1$
Conductor $2240$
Sign $0.815 - 0.579i$
Analytic cond. $10.4025$
Root an. cond. $10.4025$
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  + (0.608 + 0.793i)3-s + (−0.258 + 0.965i)9-s + (0.991 + 0.130i)11-s + (−0.923 − 0.382i)13-s + (−0.866 − 0.5i)17-s + (−0.130 − 0.991i)19-s + (0.258 − 0.965i)23-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.793 + 0.608i)37-s + (−0.258 − 0.965i)39-s + (−0.707 − 0.707i)41-s + (−0.382 − 0.923i)43-s + ⋯
L(s)  = 1  + (0.608 + 0.793i)3-s + (−0.258 + 0.965i)9-s + (0.991 + 0.130i)11-s + (−0.923 − 0.382i)13-s + (−0.866 − 0.5i)17-s + (−0.130 − 0.991i)19-s + (0.258 − 0.965i)23-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.793 + 0.608i)37-s + (−0.258 − 0.965i)39-s + (−0.707 − 0.707i)41-s + (−0.382 − 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.815 - 0.579i$
Analytic conductor: \(10.4025\)
Root analytic conductor: \(10.4025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2240,\ (0:\ ),\ 0.815 - 0.579i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.570193785 - 0.5009204620i\)
\(L(\frac12)\) \(\approx\) \(1.570193785 - 0.5009204620i\)
\(L(1)\) \(\approx\) \(1.206848540 + 0.1052161929i\)
\(L(1)\) \(\approx\) \(1.206848540 + 0.1052161929i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.608 + 0.793i)T \)
11 \( 1 + (0.991 + 0.130i)T \)
13 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.130 - 0.991i)T \)
23 \( 1 + (0.258 - 0.965i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.793 + 0.608i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.991 + 0.130i)T \)
59 \( 1 + (0.130 - 0.991i)T \)
61 \( 1 + (-0.991 + 0.130i)T \)
67 \( 1 + (-0.608 - 0.793i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.965 - 0.258i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (0.965 + 0.258i)T \)
97 \( 1 - T \)
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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

−19.737393735403899070495574682812, −19.27163385435879217034844830157, −18.31471693157557124120226258503, −17.73532512405876113814388452071, −16.97598405469651424868835299460, −16.30116177709644591759825098930, −15.09525435248633382458608075475, −14.700577254666934202354263063, −13.98939360479162521297907287389, −13.28677449409660524406223152839, −12.4714219299976224914713124028, −11.9175439198610864719995713479, −11.17794805877372963326373823546, −10.03084790330976713777492440538, −9.33666719520705696054459513634, −8.61650871474707435485606686588, −7.94214559882982036219246376592, −6.95595497578197242806053149653, −6.58781151696924597340370673926, −5.611734138767355097445714964461, −4.4804648591225554804482190769, −3.65814338961706044209678486124, −2.82373284077658248635728572581, −1.81763186543154194541935030365, −1.210520394593375448791183060489, 0.50207828494286014748521878412, 2.10076848554468939079780125530, 2.65601333679076670076278215811, 3.62484832730435898721919059234, 4.61877978700764015080401912481, 4.85121511302123642743570834336, 6.19202498441710969304833794538, 6.95018923814005980377687117227, 7.86428381322884644589862081293, 8.674739061490841517951970906723, 9.34127068609543981194294710945, 9.910140904643033641451183812297, 10.76390226340740635083459486548, 11.51818630592591083656475805856, 12.26703973736733099895227942701, 13.369159061791596339386284094734, 13.81040356803534182914354091195, 14.81889460892573698826461275438, 15.149232229662038981743615277495, 15.86743543234500967525888452382, 16.94142473628448203515997789930, 17.13542076385555090629942853011, 18.20692838103149436405673364479, 19.15657994866913424750891493096, 19.784115670542523308823707804451

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