Properties

Label 1-1480-1480.819-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.464 + 0.885i$
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 + (0.984 + 0.173i)13-s + (0.984 − 0.173i)17-s + (−0.642 + 0.766i)19-s + (−0.939 − 0.342i)21-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)27-s + (0.866 − 0.5i)29-s i·31-s + (−0.173 + 0.984i)33-s + (0.642 + 0.766i)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 + (0.984 + 0.173i)13-s + (0.984 − 0.173i)17-s + (−0.642 + 0.766i)19-s + (−0.939 − 0.342i)21-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)27-s + (0.866 − 0.5i)29-s i·31-s + (−0.173 + 0.984i)33-s + (0.642 + 0.766i)39-s + (−0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.464 + 0.885i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.464 + 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9342302990 + 1.545271742i\)
\(L(\frac12)\) \(\approx\) \(0.9342302990 + 1.545271742i\)
\(L(1)\) \(\approx\) \(1.161607399 + 0.5655819355i\)
\(L(1)\) \(\approx\) \(1.161607399 + 0.5655819355i\)

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 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (0.342 + 0.939i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
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

−20.25899486828795764569001554455, −19.50453302151158883623136192900, −19.124667996520298177982453962669, −18.34675058514875075635480051073, −17.5099246254325101199671130595, −16.576416338009705660892728091801, −15.91473568230700739119464280386, −15.07753373610920134985219072103, −14.13505099249015537457324974191, −13.59437159430963485194979251515, −12.98449027045627563523935504386, −12.18247977012723082441026267712, −11.30581100996580194085021148498, −10.29952822355894673124252214482, −9.49675133318015359985700320761, −8.69533228034509766210487735437, −8.073463915850848605487269903836, −7.10340192393961359934422882503, −6.32601775033463411147399098523, −5.776641227167471911490736491726, −4.137360223673276138690365741087, −3.47486274368287342354092628212, −2.81346863022578818162310112750, −1.560555876069332476881324291628, −0.62814292893950161542303913719, 1.45043506274909836273944032869, 2.43788146101812797742153745203, 3.41051106590186253524895665711, 3.98480901680227802708933016166, 4.945040869200192012590552037710, 6.0648169285282797825815904273, 6.74541967293894844062850983715, 7.93178090235915225178906821231, 8.54439162090927420218892440146, 9.44951271395124600930159940801, 9.99378105561411727143694173612, 10.64845300646466984890755684759, 11.90125987224462380407262992319, 12.542519750859134193790600502857, 13.4027640080198465222019450767, 14.273271549800095602344885149135, 14.76571530557923607247482481508, 15.80625254546653424998655538820, 16.128048568942217779856139013798, 16.94431404229819945468822999737, 18.02713083256467012673279750818, 18.90294191025478928081937458537, 19.39338210020558534095835883604, 20.23462156700818571138913979193, 20.852337870395979885361926071457

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