Properties

Label 1-2960-2960.2723-r1-0-0
Degree $1$
Conductor $2960$
Sign $-0.598 + 0.801i$
Analytic cond. $318.096$
Root an. cond. $318.096$
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 + (−0.342 + 0.939i)7-s + (0.173 − 0.984i)9-s + (0.866 + 0.5i)11-s + (0.984 − 0.173i)13-s + (0.173 − 0.984i)17-s + (−0.766 + 0.642i)19-s + (−0.342 − 0.939i)21-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)27-s + (0.5 − 0.866i)29-s i·31-s + (−0.984 + 0.173i)33-s + (−0.642 + 0.766i)39-s + (0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.342 + 0.939i)7-s + (0.173 − 0.984i)9-s + (0.866 + 0.5i)11-s + (0.984 − 0.173i)13-s + (0.173 − 0.984i)17-s + (−0.766 + 0.642i)19-s + (−0.342 − 0.939i)21-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)27-s + (0.5 − 0.866i)29-s i·31-s + (−0.984 + 0.173i)33-s + (−0.642 + 0.766i)39-s + (0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.598 + 0.801i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ -0.598 + 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7217255981 + 1.439165354i\)
\(L(\frac12)\) \(\approx\) \(0.7217255981 + 1.439165354i\)
\(L(1)\) \(\approx\) \(0.8375608066 + 0.3397223183i\)
\(L(1)\) \(\approx\) \(0.8375608066 + 0.3397223183i\)

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 \( 1 + (-0.342 + 0.939i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
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 \( 1 - iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 - 0.5i)T \)
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 \( 1 - iT \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (0.342 + 0.939i)T \)
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

−18.851844303999679317250071714012, −17.83636165344321362214828922171, −17.21074166759266262208501806415, −16.71390533544303442287901421038, −16.1741477006350789809655505345, −15.22161757820592240863479116582, −14.18709524973572168927864150760, −13.74964776679626304155719010577, −12.81700803620847849005694820425, −12.571893704967659138211391672869, −11.39082081009796400319990634542, −10.91441631460210552866160591618, −10.45374781903402280564456384690, −9.32424058949451585678437719034, −8.50622267198675109531010656055, −7.79336260722827770938933797903, −6.75402027588565792683314406530, −6.46798057146211776625408790526, −5.78302033922869913798958109223, −4.59782822865369546085971963088, −4.02304314372544590792630298374, −3.08184380440813728236203827880, −1.846059452010157051073921431572, −1.062154412771328233821483500064, −0.398915217749371465939579947677, 0.81737113627221549142726589794, 1.74094556025692246120339586066, 2.96454342948533062836977696403, 3.68637342266732296368667402830, 4.52952924959540590046417842368, 5.30866652482304707885236212380, 6.0923938430800312065216746034, 6.52652043999336421914256899740, 7.529960282124226518891122284859, 8.72938445430252119895070341797, 9.14459186150585592284639236166, 9.90165280685921700411150435469, 10.61672587822491576056306042118, 11.496125324638931877887587385113, 11.97735136707985967660297236587, 12.56497712040544604153248377980, 13.50974253548045407549611547512, 14.399150210471257782942465444534, 15.23625088909863689124845713255, 15.62142576165050946729356745798, 16.35154321818522655433591778558, 17.01935563525032188474654614614, 17.754211163178343642958494794145, 18.35920223702564726704333524857, 19.052115330401944933874096665841

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