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Properties

Label 1-1480-1480.757-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.998 + 0.0536i$
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  + (−0.342 + 0.939i)3-s + (−0.984 + 0.173i)7-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.766 − 0.642i)17-s + (0.342 − 0.939i)19-s + (0.173 − 0.984i)21-s + (−0.5 − 0.866i)23-s + (0.866 − 0.5i)27-s + (0.866 + 0.5i)29-s − i·31-s + (−0.642 − 0.766i)33-s + (−0.342 − 0.939i)39-s + (−0.766 + 0.642i)41-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)3-s + (−0.984 + 0.173i)7-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.766 − 0.642i)17-s + (0.342 − 0.939i)19-s + (0.173 − 0.984i)21-s + (−0.5 − 0.866i)23-s + (0.866 − 0.5i)27-s + (0.866 + 0.5i)29-s − i·31-s + (−0.642 − 0.766i)33-s + (−0.342 − 0.939i)39-s + (−0.766 + 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.998 + 0.0536i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.998 + 0.0536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6766291143 + 0.01817324453i\)
\(L(\frac12)\) \(\approx\) \(0.6766291143 + 0.01817324453i\)
\(L(1)\) \(\approx\) \(0.6751026271 + 0.1853353447i\)
\(L(1)\) \(\approx\) \(0.6751026271 + 0.1853353447i\)

Euler product

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

−20.43382083633614823997855676743, −19.61634937555518185891947693547, −19.27473150492634713839587131762, −18.4365515743237795565070149455, −17.666053068917220646879632162089, −17.01774551391474854290807663199, −16.18906599818938148128293794195, −15.55961276556155146864037185191, −14.42258284673187425115011850282, −13.55571632155444808209081059203, −13.14526484730150768317064457105, −12.27256039098928063872504544810, −11.72958310827930698562617949001, −10.53707824482037308803063445383, −10.15276619318987606197371753492, −8.88445485846697100275906110793, −8.104694354438654269277859514739, −7.36820770920556284328239349366, −6.49410163812652383189699887844, −5.84331630802607798416939866603, −5.09154269796198412231218770281, −3.6736265832457869408417139415, −2.90701369093764546288360006922, −1.938010448736372812237147305380, −0.70579309738428354500340588536, 0.392994582523189109482376690651, 2.33129387692638129814244385435, 2.89828679885030068503186682982, 4.142926081166874346168470009296, 4.72118847204086200607297943288, 5.54337154155629181402307590285, 6.65618768673664775783127736025, 7.10384562896410182092049495178, 8.52764146884903872021909454772, 9.27284103354954309346494425683, 9.92374950489175801006899996805, 10.44901232886793136139287124425, 11.6032957274196520894161528034, 12.0790541678089753250898535626, 13.059418776059549802467035108578, 13.84638339780926957307488744241, 14.957886945909436778061943489433, 15.35952418500073720817260790106, 16.2500957897714354295013167778, 16.64948254407247860117329502162, 17.679155175105344723342816294323, 18.2403376466720315827569388871, 19.317058649703958013140425953824, 20.12252865472542429082808141450, 20.51132111922333498290806990716

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