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Properties

Label 1-1480-1480.349-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.957 - 0.287i$
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  + (0.766 − 0.642i)3-s + (0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.173 + 0.984i)17-s + (−0.766 + 0.642i)19-s + (0.939 − 0.342i)21-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)27-s + (0.5 − 0.866i)29-s + 31-s + (−0.173 − 0.984i)33-s + (0.766 + 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.173 + 0.984i)17-s + (−0.766 + 0.642i)19-s + (0.939 − 0.342i)21-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)27-s + (0.5 − 0.866i)29-s + 31-s + (−0.173 − 0.984i)33-s + (0.766 + 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.476790110 - 0.3633418102i\)
\(L(\frac12)\) \(\approx\) \(2.476790110 - 0.3633418102i\)
\(L(1)\) \(\approx\) \(1.575430788 - 0.2165614673i\)
\(L(1)\) \(\approx\) \(1.575430788 - 0.2165614673i\)

Euler product

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

−20.7471109717978366527779463742, −20.071574889171137166295125985484, −19.462415876939581884165372040, −18.41300664330981084140629544875, −17.61382267440683771962383269914, −17.03876794930419814371036926017, −15.980517691075087286054849215087, −15.31959841806202894136765100291, −14.65845851436376168561950661776, −14.08293138578760826383922895387, −13.238165818684584916286279622185, −12.37929362349083112228115519556, −11.3402808865877768101089405970, −10.57914232546387537386128440712, −10.01813497452499278643703496913, −8.91136364732647401291393496887, −8.49945619529081060130092359883, −7.47775363680237303488501441490, −6.879975750981367194507105513882, −5.42668597036781175556967069218, −4.64100341696936448557374010035, −4.147733021029146495672853514769, −2.91590665775762060075667840992, −2.23390625490032518044403929693, −1.01408561337545733037458515624, 1.19291056394942342350515541388, 1.815123149084908773974832214028, 2.794429012465289910284693950240, 3.85791374104474960426779437402, 4.542174793281134884004464199019, 6.01194985116746780914059863244, 6.37689949233969688310920194668, 7.58788321236595042291034227890, 8.25309009997029687892604562021, 8.81539842662337541077840228774, 9.587182781606735520321846290721, 10.814183498468298785113000770030, 11.55969892699090813774638622581, 12.21561233676461951617895148051, 13.14593148661367082447212029759, 13.93499678111865939227546299347, 14.440182084894573920527642598972, 15.141505200548636118908180181739, 15.99247527427402833137314554199, 17.21122273247521208102757564657, 17.49755623112197557233056427166, 18.70303221354965486919439635199, 19.07064013856768168911045396987, 19.64056995432095245700201878379, 20.78104791131012701932160941075

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